FREDHOLM THEORY FOR DEGENERATE PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

WITH FIBRED BOUNDARIES

Robert Lauter

FB 17-Mathematik, Universität Mainz D-55099 Mainz, Germany e-mail: lauter@mathematik.uni-mainz.de

and

Sergiu Moroianu

Institutul de Matematica al Academiei Romane, PO. Box 1-764, RO-70700, Bucharest, Romania e-mail: Sergiu.Moroianu@imar.ro

Abstract

We consider the calculus Ψ_{de}^{∗,∗}(X,^{de}Ω^{\frac{1}{2}} ) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects.

We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ_{de}^{∗,∗}(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah-Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ^{∗}-algebra, hence its K-theory coincides with that of its C^{∗}-closure, and we give a description of the corresponding cyclic 6-term exact sequence.

We define a Wodzicki-type residue trace on an ideal in Ψ_{de}^{∗,∗}(X,^{de}Ω^{\frac{1}{2}} ), and we show that it coincides with Dixmier’s trace for operators of order − dim X in this ideal. This extends a result of Connes for the closed case.

1 Introduction

In contrast to the case of closed manifolds there is still no satisfying general theory for the analysis of elliptic partial differential equations on open or singular manifolds. In many cases, however, desingularization or compactification leads to compact manifolds with boundary or corners, where the boundary might be equipped with an additional structure. The original problem transforms to a (degenerate) elliptic problem on the compact manifold ([31], [32], [40], [41]).

It has been proposed by R. B. Melrose ([28], [29], [33]) to use Lie subalgebras V_{B} of the Lie algebra of all vector fields V to distinguish among different degeneracies at the boundary. Clearly, the enveloping algebra Diff_{B}^{∗} of V_{B} is then the algebra of all differential operators having the degeneracy described by V_{B}. As in the closed case it is desirable to embed Diff_{B}^{∗} into a calculus of pseudodifferential operators in which elliptic operators could be inverted up to residual operators that are negligible for the calculus.

As described in [33], the question of the existence of such a pseudodifferential calculus can be reduced to the construction of certain compact manifolds with corners and maps between them. Let us call this structure for the moment a boundary fibration structure. Note that for a given manifold with boundary there are always lots of different boundary fibration structures.

In the present paper, we consider a boundary fibration structure that is associated to a compact manifold whose boundary is the total space of a fibration π : ∂X ⟶ Y of closed manifolds. We are going to describe the construction of the pseudodifferential calculus associated to this boundary fibration structure in detail. A first choice for a Lie subalgebra of V(X) is the Lie algebra V_{e}(X) of all vector fields that are tangent to the fibers of π at the boundary. The corresponding boundary fibration structure is called the edge structure; it has been considered in detail in [25], and in a slightly different form for instance also in [37] or [40]. The analysis of the corresponding edge calculus is beyond the scope of our work.

Instead we concentrate on the double-edge calculus; its Lie algebra of double-edge vector fields is given by V_{de}(X) = ϱ_{N}V_{e}(X) where ϱ_{N} : X ⟶ \overline{ℝ}_{+} is a boundary defining function, i.e. ∂X = {ϱ_{N} = 0} and dϱ_{N} ≠ 0 at ∂X. The double-edge structure was first considered in [33]. If (x, y, z) ∈ \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m} are local product coordinates near the boundary, a typical double-edge vector field is a linear combination of x^{2}∂_{x}, x^{2}∂_{y}, and x∂_{z}. Though, at the first glance, these vector fields, and hence also the corresponding double-edge differential operators, look even more complicated than the edge vector fields x∂_{x}, x∂_{y}, and ∂_{z}, it turns out that the double-edge calculus is very similar to the usual pseudodifferential calculus on closed manifolds. This is essentially a consequence of the fact that the quotient Lie algebra V_{de}(X)/ϱ_{N}V_{de}(X) is commutative, or more precisely, a bundle of commutative Lie algebras over the boundary, whereas the corresponding Lie algebra V_{e}(X)/ϱ_{N}V_{e}(X) in the edge case is non-commutative. Let us elaborate this point a little. Note that there are two interesting filtrations on the algebra of edge resp. double-edge differential operators; the filtration given by the operator order and the filtration given by the order of vanishing at the boundary. Exactly as in the closed case, edge as well as double-edge differential operators commute up to lower order terms in the order filtration. The commutativity of V_{de}(X)/ϱ_{N}V_{de}(X), however, implies that double-edge differential operators commute up to lower order terms in the boundary filtration as well, i.e. the associated graded algebra is commutative. The same will hold for double-edge pseudodifferential operators. This explains the similarity of the double-edge calculus with the calculus on closed manifolds. In contrast to that, the non-commutativity of V_{e}(X)/ϱ_{N}V_{e}(X) leads to non-local, non-commutative terms in the boundary algebra making the edge calculus completely different.

Let us mention two special cases of the double-edge structure. If the fibers of the fibration are points (no z’s), then the corresponding boundary fibration structure is also known as double-zero structure [33], whereas in the case when the base Y of the fibration is a point (no y’s) the boundary fibration structure is called the scattering structure ([31], [32]).

A third boundary structure on X is given by the Lie algebra of all edge vector fields V satisfying V ϱ_{N} ∈ ϱ_{N}^{2}C^{∞}(X). This fibered-cusp structure depends on the choice of the boundary defining function ϱ_{N} and has been considered in [26].

Let us now give a more detailed plan of the paper: in Section 2 we discuss the three basic objects that are needed to define double-edge pseudodifferential operators, namely the double-edge tangent bundle ^{de}T X, and the double-edge double and triple spaces X_{de}^{2} and X_{de}^{3}. The main tool used for the construction of the manifolds with corners X_{de}^{2} and X_{de}^{3} is blowing up p-submanifolds of manifolds with corners. These techniques were developed in [28] in full generality; for a brief review see [9], [27], or [31].

Section 3 is devoted to the definition of the calculus Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) of double-edge pseudodifferential operators and the description of its symbolic structure. Here m ∈ ℝ indicates the pseudodifferential order, and k ∈ ℝ measures the order of vanishing to the boundary. By a double-edge quantization we reduce the analysis of operators to the analysis of formal series of “smooth” functions equipped with a star-product up to smoothing remainders. The leading term of this series gives the essential tool for the elliptic theory of double-edge pseudodifferential operators, the joint double-edge symbol map τ_{de}^{(m,k)} inducing the following short exact sequence

0 ⟶ Ψ_{de}^{m−1,k−1}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶^{τ_{de}^{(m,k)}} C^{∞}(C_{de}X) ⟶ 0 .

The important point to note here is that the range of the joint double-edge symbol is a commutative algebra of functions.

The symbolic calculus for double-edge operators is used in Section 4 to construct a scale of (weighted) Sobolev spaces ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) on which the double-edge operators act naturally. In particular, we show the boundedness of Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) on weighted L^{2}-spaces using the symbolic approach of Hörmander [16].

In Section 5 we characterize those double-edge operators that are Fredholm or compact using the joint double-edge symbol map; more precisely, a double-edge pseudodifferential operator a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) is Fredholm resp. compact as an operator ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) if and only if τ_{de}^{(m,k)}(a)(ω) ≠ 0 resp. τ_{de}^{(m,k)}(a)(ω) = 0 for all ω ∈ C_{de}X. We present a formula for the index of a Fredholm operator in Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) in terms of the K-class given by the joint double-edge symbol [τ_{de}^{(m,k)}(a)]. This formula is in the spirit of the classical formula for closed manifolds of Atiyah and Singer [1]. Moreover, we prove the existence of a double-edge operator of index 1.

In Section 6 we compute the K-theory of the C^{∗}-algebra B(X,^{de}Ω^{\frac{1}{2}} ) generated by the algebra Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) of double-edge operators of order (0, 0). The algebra Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) itself turns out to be a Ψ^{∗}-algebra in B(X,^{de}Ω^{\frac{1}{2}} ) in the sense of B. Gramsch [13, Definition 5.1]. This has a few remarkable consequences, for instance for the structure of the set of idempotent or relatively regular elements in Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ); for more details and applications we refer the reader to the brief review in [21, Remark 2.10], and the references given therein. Let us only mention that Ψ^{∗}-algebras are stable under the (multidimensional) holomorphic functional calculus of J. L. Taylor [19], in particular, the K-Theory of Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) coincides with that of B(X,^{de}Ω^{\frac{1}{2}} ).

In the last Section 7 we show that the space Ψ_{de}^{− dim X,−k}(X,^{de}Ω^{\frac{1}{2}} ) is included in the Dixmier ideal ℓ^{1,∞} for any k > 2 dim X; in particular, the Dixmier traces Tr_{ω} are well-defined on Ψ_{de}^{− dim X,−k}(X,^{de}Ω^{\frac{1}{2}} ). Extending a result of A. Connes [6] from the closed to the double-edge case we show that up to a universal constant Tr_{ω}(a) = C_{N}W-Res_{de}(a) for all a ∈ Ψ_{de}^{− dim X,−k}(X,^{de}Ω^{\frac{1}{2}} ); here W-Res_{de} is the extension of the double-edge Wodzicki-residue, or more precisely, of the unique trace on Ψ_{de}^{ℤ,−∞}(X,^{de}Ω^{\frac{1}{2}} ) := ⋃_{m∈ℤ} Ψ_{de}^{m,−∞}(X,^{de}Ω^{\frac{1}{2}} ) that vanishes on Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ), to Ψ_{de}^{ℤ,−k}(X,^{de}Ω^{\frac{1}{2}} ). It is worth pointing out that the Dixmier traces replace the integral in Connes’ formulation of non-commutative geometry [7]. A similar result for boundary value problems has been obtained in [35] – see also [11], [38], and [39].

Finally, let us mention that we compute in [22] the Hochschild homology and describe all traces for various ideals and quotients of Ψ_{de}^{∗,∗}(X,^{de}Ω^{\frac{1}{2}} ).

To fix some notations, let E ⟶ Z be a smooth vector bundle over a manifold Z with corners. Then Ċ^{∞}(Z, E) stands for the space of smooth sections Z ⟶ E of E vanishing to infinite order at the boundary of Z. Moreover, S^{[m]}(E) denotes the space of all smooth functions f : E \ {0} ⟶ ℂ homogeneous of degree m ∈ ℝ in the radial direction, i.e. f(λv) = λ^{m}f(v) for λ > 0 and v ∈ E \ {0}.

2 Geometric constructions in the double-edge

category

In this section we introduce the basic geometric objects underlying the doubleedge calculus. Let X denote a smooth, compact manifold whose boundary ∂X is the total space of a (locally trivial) fiber bundle π : ∂X ⟶ Y, where the fiber F^{m} as well as the base Y^{n} are closed manifolds. Let

N := dim X = dim F + dim Y + 1 = m + n + 1

be the dimension of X. For simplicity, we will assume that the boundary ∂X is connected.

The double-edge structure space

Recall [25] that a smooth vector field v ∈ C^{∞}(X, T X) is said to be an edge vector field if it is tangent to the fibers of π at the boundary. Let us denote by V_{e}(X) the Lie algebra of all edge vector fields on X. Moreover, choose a boundary defining function ϱ_{N} : X ⟶ \overline{ℝ}_{+} for X, i.e. a smooth function with ∂X = {ϱ_{N} = 0} and dϱ_{N} ≠ 0 at the boundary. 2.1 Definition. The elements of V_{de}(X) := ϱ_{N}V_{e}(X) are called double-edge vector fields.

Note that the set of double-edge vector fields does not depend on the choice of the boundary defining function ϱ_{N}. As usual, double-edge vector fields form a sheaf of C^{∞}(X)-modules. To give a local description of edge resp. double-edge vector fields choose a local product decomposition

(x, y, z) : X ⊇ U ⟶ \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}  (1)

of X near the boundary such that ϱ_{N}|_{U} = x, y|_{U∩∂X} restricts to a coordinate system that is lifted from Y, and z|_{U∩∂X} to one on the fibers. 2.2 Proposition.

(a) The space V_{e}(X)|_{U} is spanned over C^{∞}(U) by the vector fields x∂_{x}|_{U}, x∂_{y_{j}}|_{U}, j = 1, …, n, and ∂_{z_{k}}|_{U}, k = 1, …, m.

(b) The space V_{de}(X)|_{U} is spanned over C^{∞}(U) by the vector fields x^{2}∂_{x}|_{U}, x^{2}∂_{y_{j}}|_{U}, j = 1, …, n, and x∂_{z_{k}}|_{U}, k = 1, …, m.

As usual these basic edge resp. double-edge vector fields can be used to define the corresponding structure bundles. 2.3 Proposition. There exist smooth vector bundles ^{e}T X ⟶ X resp. deT X ⟶ X together with natural vector bundle morphisms j^{e} : ^{e}T X ⟶ T X resp. j^{de} : ^{de}T X ⟶ T X such that we have V_{e}(X) = j^{e}(C^{∞}(X,^{e}T X)) and V_{e}(X) = j^{de}(C^{∞}(X,^{de}T X)).

The bundles ^{e}T X resp. ^{de}T X are called the edge resp. double-edge tangent bundle. From now on, we restrict ourselves mostly to the double-edge case. 2.4 Proposition. The space V_{de}(X) of all double-edge vector fields is a Lie subalgebra of the Lie algebra V(X) of all vector fields and satisfies [V_{de}(X), V_{de}(X)] ⊆ ϱ_{N}V_{de}(X). In particular, the quotient V_{de}(X)/ϱ_{N}V_{de}(X) is a commutative Lie algebra.

Note that we have [V_{de}(X), ϱ_{N}V_{de}(X)] ⊆ ϱ_{N}V_{de}(X), i.e. ϱ_{N}V_{de}(X) is an ideal in V_{de}(X), hence the quotient algebra is well-defined. Proof: This can be easily checked using the description in local coordinates given in Proposition 2.2. □

It is worth pointing out that the corresponding quotient Lie algebra V_{e}(X)/ϱ_{N}V_{e}(X) in the edge setting is not commuative. This is, in fact, the main difference between the edge and the double-edge case. Let ^{de}T^{∗}X ⟶ X be the double-edge cotangent bundle. Since φ_{de} := (j^{de})^{∗} : T^{∗}X ⟶ ^{de}T^{∗}X is an isomorphism if restricted to the interior X_{0} := X \ ∂X, the symplectic form ω on T^{∗}X can be pushed forward to a closed 2-form ω_{de} on ^{de}T^{∗}X|_{X_{0}} giving^{de}T^{∗}X|_{X_{0}} the structure of a symplectic manifold. Using local coordinates (x, y, z) near the boundary, and the associated coordinates (x, y, z, ξ, η, ζ) on deT^{∗}X, ω_{de} is given by

ω_{de} = \frac{dx}{x^{2}} ∧ dξ + \frac{dy}{x^{2}} ∧ dη + \frac{dz}{x} ∧ dζ + 2xη \frac{dx}{x^{2}} ∧ \frac{dy}{x^{2}} + xζ \frac{dx}{x^{2}} ∧ \frac{dz}{x} , (2)

i.e. we have ω_{de} ∈ C^{∞}(^{de}T^{∗}X, Λ^{2}(^{de}T^{∗}^{de}T^{∗}X)). The associated double-edge Poisson bracket {f, g}_{de} is locally given by

{f, g}_{de}
= (x^{2}∂_{x}f∂_{ξ}g − x^{2}∂_{x}g∂_{ξ}f) + (x^{2}∂_{y}f∂_{η}g − x^{2}∂_{y}g∂_{η}f) + (x∂_{z}f∂_{ζ}g − x∂_{z}g∂_{ζ}f)
+ x(2η∂_{η}f∂_{ξ}g − 2η∂_{η}g∂_{ξ}f) + x(ζ∂_{ζ}f∂_{ξ}g − ζ∂_{ζ}g∂_{ξ}f) .  (3)

Let RC : ^{de}T^{∗}X ⟶ ^{de}\overline{T}^{∗}X be the radial compactification of the doubleedge cotangent bundle ^{de}T^{∗}X as in [31, Section 1], and

deS^{∗}X := ^{de}T^{∗}X \ \{0\}/ℝ_{+} ⊆ ^{de}\overline{T}^{∗}X

be the sphere bundle at infinity, the so-called double-edge cosphere bundle. Similary, we denote by RC : ^{de}T X ⟶ ^{de}\overline{T} X the radial compactification of the double-edge tangent bundle.

The double-edge double space X_{de}^{2}

Double-edge pseudodifferential operators on X will be defined by characterizing the singularities of their Schwartz kernels on the manifold with corners X^{2}. A relatively simple description of these singularities can be given on a slightly more complicated space, the so-called double-edge double space X_{de}^{2}. It is a compact manifold with corners together with a surjective blowdown map β_{de}^{2} : X_{de}^{2} ⟶ X^{2} that induces an isomorphism between the spaces C^{−∞}(X_{de}^{2}, Ω^{\frac{1}{2}} ) and C^{−∞}(X^{2}, Ω^{\frac{1}{2}} ) of extendible distributions. For the construction of the space X_{de}^{2} we use the general concept of blowing up p-submanifolds as developed in [28]. For a quick review of blow-up techniques see for instance [27], [30], or [31, Section 18].

Let B_{1} := {(p, q) ∈ (∂X)^{2} : π(p) = π(q)} be the fiber-diagonal at the boundary. Then

β_{e}^{2} : X_{e}^{2} := [X^{2}; B_{1}] ⟶ X^{2}  (4)

is known as the edge double space [25, Section 2]. We call the submanifold Δ_{e} := \overline{(β_{e}^{2})^{−1}(Δ \ (∂X)^{2})}^{X_{e}^{2}} the lifted diagonal. The new boundary hypersurface of X_{e}^{2} produced by the blow up is called the edge front face ff^{e}. Note that, by definition of the blow up, ff^{e} is a fiber bundle over B_{1}, hence a fiber bundle over Y. Let us denote by

π_{ff}^{e} : ff^{e} ⟶ B_{1} ⟶^{π} Y  (5)

the projection map of this fiber bundle.

For the second blow-up let B_{2} := ∂X_{e}^{2} ∩ Δ_{e} = ff^{e} ∩ Δ_{e} be the boundary of the lifted diagonal. The double-edge blow-up is then given as the composition

β_{de}^{2} : X_{de}^{2} := [X_{e}^{2}; B_{2}] ⟶^{β} X_{e}^{2} ⟶^{β_{e}^{2}} X^{2} .  (6)

As above, let Δ_{de} := \overline{β^{−1}(Δ_{e} \ ∂X_{e}^{2})}^{X_{de}^{2}} be the lifted diagonal, and denote by ff^{de} the double-edge front face, i.e. the new boundary hypersurface of X_{de}^{2} produced by the blow up. Let ϱ_{ff}_{de} : X_{de}^{2} ⟶ \overline{ℝ}_{+} be a defining function of the double-edge front face.

For a local description of X_{de}^{2} near the front face ff^{de} choose local product coordinates (x, y, z) as in (1), and denote by (x, y, z) resp. (x′, y′, z′) the lift of these coordinates through the projection onto the left resp. right factor of X^{2}. In most computations we will use one of the following sets of projective coordinates near the front face. The first system is given by

(S, U, Z, x′, y′, z′) where S = \frac{x − x′}{x^{′2}} , U = \frac{y − y′}{x^{′2}} , and Z = \frac{z − z′}{x′} , (7)

whereas for the second set of projective coordinates we have

(T, U, W, x, y, z) with T = \frac{x′ − x}{x^{2}}, V = \frac{y′ − y}{x^{2}} , and W = \frac{z′ − z}{x} . (8)

With respect to these coordinates the lifted diagonal Δ_{de} and the double-edge front face ff^{de} are given by

Δ_{de} = {S = 0, U = 0, Z = 0} = {T = 0, V = 0, W = 0} and  (9)
ff^{de} = {x′ = 0} = {x = 0} .  (10)

Note that a coordinate neighborhood near the interior of the front face ff^{de} is given by

{(S, U, Z, x′, y′, z′) : 1 + x′S > 0, 0 ≤ x′ < ε} .  (11)

2.5 Proposition. The Lie algebra V_{de}(X) of double-edge vector fields lifts, from either the left or right factor of X^{2}, to a Lie subalgebra of V_{b}(X_{de}^{2}). The lifted vector fields are transversal to the lifted diagonal Δ_{de}. Proof: Since the blow down map β_{de}^{2} : X_{de}^{2} ⟶ X^{2} restricted to the interior of X_{de}^{2} is a diffeomorphism, and all constructions are local, it suffices to consider local coordinates. We restrict ourselves to the product decomposition given by (1) and the local coordinates (7) near the front face. The other cases are similar. A straightforward computation using (7) shows then that the basic vector fields of Proposition 2.2 lift through the projection onto the left factor of X^{2} to

x^{2}∂_{x} ⇝ (1 + x′S)^{2}∂_{S}
x^{2}∂_{y} ⇝ (1 + x′S)^{2}∂_{U}
x∂_{z} ⇝ (1 + x′S)∂_{Z}

on X_{de}^{2} which completes the proof in this case because of (9) and (10). The lift through the projection onto the right factor follows by symmetry. □

Note that the composition of the blow-down map β_{de}^{2} : X_{de}^{2} ⟶ X^{2} with the projection X^{2} ⟶ X onto either the left or the right factor restricts to a canonical diffeomorphism Δ_{de} ⟶ X; the transversality of the lift of double-edge vector fields to Δ_{de} shows that there are natural isomorphisms

\begin{aligned}^{de}T X & ⟶^{≅} & NΔ_{e} \\ ↓ & & ↓ \\ X & ⟶^{≅} & Δ_{de} \end{aligned} and \begin{aligned}N^{∗}Δ_{de} & ⟶^{≅} & ^{de}T^{∗}X \\ ↓ & & ↓ \\ Δ_{de} & ⟶^{≅} & X \end{aligned}  (12)

Moreover, integrating lifted vector fields at the front face leads to a natural identification

NΔ_{de}|_{Δ_{de}∩ff^{de}} ⟶^{≅} int ff^{de}  (13)

that extends to a diffeomorphism

de\overline{T} X|_{∂X} ⟶^{≅} ff^{de} .  (14)

Let us stress the fact that the blow down map β_{de}^{2} : X_{de}^{2} ⟶ X^{2} induces, via pull-back and duality, isomorphisms (β_{de}^{2})^{∗} : Ċ^{∞}(X^{2}, Ω^{\frac{1}{2}} ) ⟶ Ċ^{∞}(X_{de}^{2}, Ω^{\frac{1}{2}} ), and (β_{de}^{2})_{∗} : C^{−∞}(X_{de}^{2}, Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X^{2}, Ω^{\frac{1}{2}} ). However, because of

C^{∞}(X^{2}, Ω^{\frac{1}{2}} ) ⊊ (β_{de}^{2})_{∗}(ϱff^{de}^{\frac{2+2n+m}{2}} C^{∞}(X_{de}^{2}, Ω^{\frac{1}{2}} ))

there are “more” smooth densities on X_{de}^{2} than on X^{2}. This is the flexibility needed for a simple description of the Schwartz kernels of double-edge pseudodifferential operators.

The double-edge triple space X_{de}^{3}

To show that the calculus of double-edge pseudodifferential operators is closed under composition, it is convenient to construct a “triple” version X_{de}^{3} of the double-edge double space X_{de}^{2}. We are going to follow the presentation in [31, Section 23] – see also [33] for a general approach. In fact, we extend the construction in [31, Section 23] from the scattering calculus to the double-edge calculus. We will use the following projections:

π_{C} : X^{3} ⟶ X^{2} : (q_{1}, q_{2}, q_{3}) ⟼ (q_{1}, q_{3}) ,
π_{F} : X^{3} ⟶ X^{2} : (q_{1}, q_{2}, q_{3}) ⟼ (q_{2}, q_{3}) , and
π_{S} : X^{3} ⟶ X^{2} : (q_{1}, q_{2}, q_{3}) ⟼ (q_{1}, q_{2}) .

For O ∈ {C, F, S} let B_{1,O} := π_{O}^{−1}(B_{1}) ⊆ X^{3} be the lift of the fiber diagonal under the projection π_{O}, and

B_{1,T} := {(p_{1}, p_{2}, p_{3}) ∈ (∂X)^{3} : π(p_{1}) = π(p_{2}) = π(p_{3})}

be the triple fiber diagonal in (∂X)^{3}. It is straightforward to check that the spaces B_{1,O}, O = C, F, S, T are p-submanifolds of the manifold with corners X^{3} satisfying

B_{1,T} = B_{1,O} ∩ B_{1,O′} for all O, O′ ∈ {C, F, S} with O ≠ O′ .

Let β_{e}^{3} : X_{e}^{3} := [X^{3}; B_{1,T}; B_{1,C}; B_{1,F}; B_{1,S}] ⟶ X^{3} be the composition of the corresponding blow-down maps. Note that X_{e}^{3} is exactly the edge triple space as constructed in [25, (3.12)]. Let Δ_{O} := π_{O}^{−1}(Δ) ⊆ X^{3}, O = C, F, S, resp. Δ_{T} := {(q_{1}, q_{2}, q_{3}) ∈ X^{3} : q_{1} = q_{2} = q_{3}} be the partial resp. triple diagonal in X^{3}. Then the p-submanifolds Δ_{O} lift under the blow-down map β_{e}^{3} to p-submanifolds Δ_{O,e} := (β_{e}^{3})^{∗}(Δ_{O}) of the manifold X_{e}^{3}, O = C, F, S, satisfying

Δ_{O,e} ∩ Δ_{O′,e} = Δ_{T,e} for all O, O′ ∈ {C, F, S} with O ≠ O′ ,

where Δ_{T,e} := (β_{e}^{3})^{∗}(Δ_{Y}) is the lift of the triple diagonal. Moreover, for all O, O′ ∈ {C, F, S} with O ≠ O′ the projection π_{O′,e} induces a diffeomorphism

π_{O′,e}|_{Δ_{O,e}} : Δ_{O,e} ⟶ X_{e}^{2} .

In particular, Δ_{O,e} has three boundary hypersurfaces. Let us denote by B_{2,O} that boundary hypersurface of Δ_{O,e} contained in the lift of the front face of [X^{3}; B_{1,T}] to X_{e}^{3}, and by B_{3,O} that contained in the face of X_{e}^{3} produced by blowing up B_{1,O}.

Observe that the intersection of any two of the B_{2,O}’s is the boundary of the lifted triple diagonal which we denote by

B_{2,T} := B_{2,O} ∩ B_{2,O′} = ∂Δ_{T,e} for all O, O′ ∈ {C, F, S} with O ≠ O′ .

Further note that the B_{3,O}’s are disjoint but meet the corresponding B_{2,O} away from B_{2,T}, i.e. we have

B_{3,O} ∩ B_{3,O′} = ∅ for all O, O′ ∈ {C, F, S} with O ≠ O′ ,  (15)
B_{3,O} ∩ B_{2,O′} = ∅ for all O, O′ ∈ {C, F, S} with O ≠ O′ , and (16)
B_{3,O} ∩ B_{2,T} = ∅ for all O ∈ {C, F, S} .  (17)

We can now define the double-edge triple space X_{de}^{3} by

β_{de}^{3} : X_{de}^{3} := [X_{e}^{3}; B_{2,T}; B_{2,C}; B_{2,F}; B_{2,S}; B_{3,C}; B_{3,F}; B_{3,S}] ⟶ X^{3} ,  (18)

where β_{de}^{3} is the composition of the obvious blow down map X_{de}^{3} ⟶ X_{e}^{3} with β_{e}^{3} : X_{e}^{3} ⟶ X^{3}.

The main properties of the double-edge triple space are collected in the following proposition. 2.6 Proposition. The double-edge triple space X_{de}^{3} is a compact manifold with corners such that:

(a) The action of the symmetry group in three variables acting on the three factors of X^{3} lifts to diffeomorphisms of X_{de}^{3}.

(b) There exist b-fibrations π_{O,de} : X_{de}^{3} ⟶ X_{de}^{2}, O = C, F, S making the following diagram commutative:

\begin{matrix}X_{de}^{3} & ⟶^{π_{O,de}} & X_{de}^{2} \\ β_{de}^{3}↓ & & ↓β_{de}^{2} \\ X^{3} & ⟶^{π_{O}} & X^{2} \end{matrix} .  (19)

(c) The double-edge diagonal Δ_{de} in X_{de}^{2} lifts under π_{O,de} to a p-submanifold Δ_{O,de} ⊆ X_{de}^{3} such that

π_{O′,de}|_{Δ_{O,de}} : Δ_{O,de} ⟶ X_{de}^{2}

is a diffeomorphism for all O, O′ ∈ {C, F, S} with O ≠ O′.

Proof: The symmetry condition (a) follows immediately from the symmetry of the construction of the space X_{de}^{3}. By this symmetry of X_{de}^{3} it suffices to construct the map π_{F,de} : X_{de}^{3} ⟶ X_{de}^{2} and to check (b) and (c). We will define π_{F,de} as the composition of the following maps:

π_{F,de} : X_{de}^{3} = [X_{e}^{3}; B_{2,T}; B_{2,F}; B_{2,C}; B_{2,S}; B_{3,F}; B_{3,C}; B_{3,S}]  (20)
⟶ [X_{e}^{3}; B_{2,T}; B_{2,F}; B_{2,C}; B_{2,S}; B_{3,F}]  (21)
≅ [X_{e}^{3}; B_{2,T}; B_{2,F}; B_{3,F}; B_{2,S}; B_{2,C}]  (22)
⟶ [X_{e}^{3}; B_{2,T}; B_{2,F}; B_{3,F}]  (23)
≅ [X^{3}; B_{1,F}; B_{1,T}; B_{1,C}; B_{1,S}; B_{2,F}; B_{2,T}; B_{3,F}]  (24)
≅ [X × X_{e}^{2}; B_{1,T}; B_{1,C}; B_{1,S}; B_{2,F}; B_{2,T}; B_{3,F}]  (25)
≅ [X × X_{e}^{2}; B_{2,F}; B_{1,T}; B_{3,F}; B_{1,C}; B_{1,S}; B_{2,T}]  (26)
⟶ [X × X_{e}^{2}; B_{2,F}; B_{1,T}; B_{3,F}]  (27)
≅ [X × X_{e}^{2}; B_{3,F}; B_{2,F}; B_{1,T}]  (28)
⟶ [X × X_{e}^{2}; X × ∂Δ_{e}]  (29)
≅ X × X_{de}^{2}  (30)
⟶ X_{de}^{2} ,  (31)

where (20) is the definition of the double-edge triple space, (21), (23), (27), and (29) are the obvious blow-down maps, (31) is the projection onto the second factor, and the diffeomorphism (30) is induced by the definition (6) of the double-edge double space.

For the diffeomorphism (22) we use B_{3,F} ∩ B_{2,C} = B_{3,F} ∩ B_{2,S} = ∅ by (16). For (24) we apply the commutativity of the blow up [28, Proposition 5.8.1] because of B_{1,T} ⊆ B_{1,F} and B_{2,T} ⊆ B_{2,F}, whereas (25) is a consequence of the equality B_{1,F} = X × {(p_{2}, p_{3}) ∈ (∂X)^{2} : π(p_{2}) = π(p_{3})} and the definition (4) of the edge double space X_{e}^{2}.

For the remaining two diffeomorphisms (26) and (28) note that after blowing down

X_{e}^{3} ≅ [X^{3}; B_{1,F}; B_{1,T}; B_{1,C}; B_{1,S}] ⟶ [X^{3}; B_{1,F}] ≅ X × X_{e}^{2}

we have

B_{1,T} = ∂X ×_{π} ff^{e} := {(p, q) ∈ ∂X × ff^{e} : π(p) = π_{ff}^{e}(q)} ,  (32)
B_{2,F} = ∂X ×_{π} ∂Δ_{e} := {(p, q) ∈ ∂X × ∂Δ_{e} : π(p) = π_{ff}^{e}(q)} , and (33)
B_{3,F} = X × ∂Δ_{e} ,  (34)

where π_{ff}^{e} : ff^{e} ⟶ Y is the fibration map described in (5). Since we have B_{2,F} ∩ B_{1,C} = B_{2,F} ∩ B_{1,S} = ∅ and B_{3,F} ∩ B_{1,C} = B_{3,F} ∩ B_{1,S} = ∅ by definition, B_{3,F} ∩ B_{2,T} = ∅ by (17), and B_{2,F} ⊆ B_{1,T} by (32) and (33), commutativity of the blow up leads to the diffeomorphism (26), and, finally, (28) is a consequence of B_{2,F} ⊆ B_{3,F} by (33) and (34), and the fact that B_{3,F} and B_{1,T} are disjoint in the space [X × X_{e}^{2}; B_{2,F}] where its intersection B_{2,F} = B_{1,T} ∩ B_{3,F} is blown up – note that B_{1,T} and B_{3,F} are transversal p-submanifolds of the hypersurface X × ff^{e} in X × X_{e}^{2}.

Since in both possible ways from X_{de}^{3} to X^{2} in (19) finally all blow ups are blown down again, and the order does not matter by the arguments above, the commutativity of (19) follows.

As in the proof of [31, Lemma 16] we see using [31, Proposition 16] that π_{F,de} is a b-fibration, and (c) follows by continuity from the interior. □

Double-edge densities

Finally, we need a density bundle that is adapted to the calculus. Apply the smooth functor Ω^{α} of α-densities to the double-edge cotangent bundle to obtain a smooth vector bundle denoted ^{de}Ω^{α}(X) ⟶ X, the bundle of double-edge αdensities. The choice of local product coordinates (x, y, z) as in (1) trivializes the bundle ^{de}Ω^{α}(X)|_{U}. A non-vanishing section is given by |\frac{dx}{x^{2}} \frac{dy}{x^{2n}} \frac{dz}{x^{m}} |^{α}.

In particular, we obtain a well-defined integral

∫_{X} : Ċ^{∞}(X,^{de}Ω^{1}) ⟶ ℂ  (35)

that does not extend naturally to C^{∞}(X,^{de}Ω^{1}) because of the factor x^{−(2+2n+m)} in the double-edge density at the boundary.

Furthermore, let ^{de}Ω^{α}(X^{2}) := π_{L}^{∗}^{de}Ω^{α}(X) ⊠ π_{R}^{∗}^{de}Ω^{α}(X) be the exterior tensor product of the pull-backs of the double-edge α-density bundles via the projections π_{L}, π_{R} : X^{2} ⟶ X onto the left resp. right factor.

On the double-edge double space X_{de}^{2} we will use the so-called doubleedge kernel α-density bundle ^{de}KD^{α} := ϱ_{ff^{de}}^{−α(2+2n+m)}Ω^{\frac{1}{2}} (X_{de}^{2}). It is completely characterized by the space of its C^{∞}-sections [30, Lemma 8.6], namely

C^{∞}(X_{de}^{2},^{de}KD^{α}) = ϱ_{ff^{de}}^{−α(2+2n+m)}C^{∞}(X_{de}^{2}, Ω^{\frac{1}{2}} (X_{de}^{2})) .

3 Double-edge pseudodifferential operators

Since double-edge pseudodifferential operators are defined by the singularities of their Schwartz kernels, let us first recall that by a version of the Schwartz kernel theorem there is a one-to-one correspondence between the continuous linear operators Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) on one hand, and the space of extendible distributions C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ) on the other hand [30, Theorem 4.18]. The kernel of an operator a : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) under this correspondence is denoted by κ_{a} ∈ C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ), and will be called the lifted Schwartz kernel of a. 3.1 Definition. A classical double-edge pseudodifferential operator of order m ∈ ℂ is a continuous linear operator a : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) whose lifted Schwartz kernel κ_{a} belongs to the space

\{κ ∈ I_{cl}^{m}(X_{de}^{2}, Δ_{de};^{de}KD^{\frac{1}{2}} ) : κ ≡ 0 at ∂X_{de}^{2} \ ff^{de}\} ⊆ C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ) ,

where ≡ means vanishing to infinite order. The space of all classical doubleedge pseudodifferential operators of order m ∈ ℝ is denoted by Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ).

For the definition and basic properties of the spaces I_{cl}^{m}(X, Y; E) of (classical) conormal distributions we refer the reader to [16], [17, Section 18.2], or [42]. In this paper, we will mainly be interested in the case m ∈ ℝ, however, many of the results remain true for m ∈ ℂ, often with almost the same proof. Let us mention that we could have also defined double-edge pseudodifferential operators using the space I^{m}(X_{de}^{2}, Δ_{de};^{de}Ω^{\frac{1}{2}} ) of conormal distributions instead of the space of classical one’s. In fact, most of the results that follow are true for this slightly larger class, too. However, since we prefer the principal symbol to be a function we restricted ourselves to the classical case, and do not even mention this point in any way in the notation. Note that the double-edge kernel half-density bundle is chosen in such a way that the lifted Schwartz kernel of the identity corresponds to a delta-section of the lifted diagonal Δ_{de} that is smooth up to the double-edge front face ff^{de}.

Basic properties of double-edge pseudodifferential operators

Let us continue by describing some basic consequences of this definition. Since the lifted Schwartz kernel κ_{a} of a ∈ Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) vanishes to infinite order at the faces different from the front face, a straightforward computation using a partition of unity and local coordinates similar to the proof of [30, Proposition 4.29] shows that we have a(Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} )) ⊆ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ), i.e. a : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) is continuous by the closed graph theorem, and composition of double-edge operators is well-defined. 3.2 Proposition. Ψ_{de}^{m_{1}}(X,^{de}Ω^{\frac{1}{2}} ) ∘ Ψ_{de}^{m_{2}}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ Ψ_{de}^{m_{1}+m_{2}}(X,^{de}Ω^{\frac{1}{2}} ). Proof: The proof is appropriately adapted from the proof of the corresponding composition formula for Θ-pseudodifferential operators in [9, Theorem 11.30]. Let a_{j} ∈ Ψ_{de}^{m_{j}}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary, and choose a smooth, non-vanishing section μ ∈ C^{∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ), thus, every κ ∈ C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ) can be written as κ = κ̂ · μ with κ̂ ∈ C^{−∞}(X_{de}^{2}). Then the lifted Schwartz kernel κ_{a_{1}a_{2}} of a_{1}a_{2} satisfies

κ̂_{a_{1}a_{2}} ·μ^{2} = (π_{C,de})_{∗} [(π_{S,de})^{∗}κ̂_{a_{1}} · (π_{F,de})^{∗}κ̂_{a_{2}} · (π_{S,de})^{∗}μ ⊗ (π_{F,de})^{∗}μ ⊗ (π_{C,de})^{∗}μ] ,

(36)

i.e. it is the push-forward of a distributional density on X_{de}^{3}. An application of [9, Proposition B7.20] gives

κ̂_{a_{1}a_{2}} · μ^{2} ∈ {κ ∈ I_{cl}^{m}(X_{de}^{2}, Δ_{de};^{de}KD^{1}) : κ ≡ 0 at ∂X_{de}^{2} \ ff^{de}} ,

and completes the proof.

For applying [9, Proposition B7.20] we first have to transform the singular density (π_{S,de})^{∗}μ ⊗ (π_{F,de})^{∗}μ ⊗ (π_{C,de})^{∗}μ into a b-density on X_{de}^{3}. This transformation leads to additional powers of defining functions of faces of X_{de}^{3}.

However, these factors can be ignored because of the rapid vanishing of the kernels at the faces other than the front face. The fact that the density factors fit together follows by taking a_{1} = id in (36). □

The integral ∫_{X} : Ċ^{∞}(X,^{de}Ω^{1}) ⟶ ℂ leads to a natural scalar product on Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ):

< f, g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}:= ∫_{X} f\overline{g} for f, g ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) .  (37)

3.3 Proposition. Let a ∈ Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Then there exists a (uniquely determined) double-edge operator a^{♯} ∈ Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) such that

< af, g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}=< f, a^{♯}g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} for all f, g ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) . (38)

The operator a^{♯} is called the formal adjoint of a. Proof: Let c : C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ) ⟶ C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ) be the operator of complex conjugation, and α : X^{2} ⟶ X^{2} be the flip. Then α extends to a diffeomorphism α : X_{de}^{2} ⟶ X_{de}^{2}, and the operator with Schwartz kernel κ := cα^{∗}(κ_{a}) satisfies (38). On the other hand, Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) is clearly invariant under c and α^{∗} which completes the proof. □

Therefore, we can extend the action of a ∈ Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) to the space C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) of extendible distributions by defining

a : C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) : u ⟼ [Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ∋ φ ⟼ u (a^{♯}(φ)] ,

and similar to [30, Proposition 4.34] we see a(C^{∞}(X,^{de}Ω^{\frac{1}{2}} )) ⊆ C^{∞}(X,^{de}Ω^{\frac{1}{2}} ), i.e. the linear operator a : C^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{∞}(X,^{de}Ω^{\frac{1}{2}} ) is continuous by the closed graph theorem.

The extended double-edge calculus

Let now ϱ_{N} : X ⟶ \overline{ℝ}_{+} be a boundary defining function, and k ∈ ℝ be arbitrary. Then multiplication with ϱ_{N}^{k} induces a bounded linear operator ϱ_{N}^{k} : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ). 3.4 Lemma. For all k ∈ ℝ we have ϱ_{N}^{−k}Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} )ϱ_{N}^{k} = Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ).

Proof: It suffices to show ϱ_{N}^{−k}Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} )ϱ_{N}^{k} ⊆ Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) for k ∈ ℝ. Let a ∈ Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Since the definition of double-edge pseudodifferential operators is invariant under change of local coordinates, we can choose local product coordinates (x, y, x) : X ⊇ U ⟶ \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m} near the boundary such that ϱ_{N}|_{U} = x : U ⟶ \overline{ℝ}_{+}. Then the lifted Schwartz kernel κ_{b} of b = ϱ_{N}^{−k}aϱ_{N}^{k} is given by

κ_{b}(S, U, Z, x′, y′, z′) = (1 + x′S)^{−k}κ_{a}(S, U, Z, x′, y′, z′)  (39)

which obviously has the desired properties. □

3.5 Definition. For k, m ∈ ℝ let

Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) := ϱ_{N}^{−k}Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) = Ψ_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} )ϱ_{N}^{−k} .

Clearly, the spaces Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) do not depend on the particular choice of the boundary defining function ϱ_{N}. Furthermore, note that we have

Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ Ψ_{de}^{m′,k′}(X,^{de}Ω^{\frac{1}{2}} ) ⟺ m′ − m ∈ ℕ_{0} and k′ − k ∈ ℕ_{0} , (40)

which was the reason for defining Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) using ϱ_{N}^{−k}. A combination of Proposition 3.2 and Lemma 3.4 gives

Ψ_{de}^{m_{1},k_{1}}(X,^{de}Ω^{\frac{1}{2}} ) ∘ Ψ_{de}^{m_{2},k_{2}}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ Ψ_{de}^{m_{1}+m_{2},k_{1}+k_{2}}(X,^{de}Ω^{\frac{1}{2}} ) ,  (41)

and for a = bϱ_{N}^{−k} ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) the operator a^{♯} := ϱ_{N}^{−k}b^{♯} ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) satisfies

< af, g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}=< f, a^{♯}g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} for all f, g ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) .

3.6 Remark. Let (S, U, Z, x′, y′, z′) be local coordinates on X_{de}^{2} near ff^{de}. Then x′ is a defining function ϱ_{ff}_{de} : X_{de}^{2} ⟶ \overline{ℝ}_{+} of the front face ff^{de} as well as the lift through the right factor of X^{2} of a boundary defining function ϱ_{N} : X ⟶ \overline{ℝ}_{+} of X; thus, the Schwartz kernel theorem yields the following identification in C^{−∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} )

Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟷^{1−1} ϱ_{ff^{de}}^{−k} \{κ ∈ I_{cl}^{m}(X_{de}^{2}, Δ_{de};^{de}KD^{\frac{1}{2}} ) : κ ≡ 0 at ∂X_{de}^{2} \ ff^{de}\} .

By (42), the elements in Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) := ⋂_{m∈ℝ} ⋂_{k∈ℝ} Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) have lifted Schwartz kernels in Ċ^{∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ), hence the Schwartz kernels themselves are in Ċ^{∞}(X^{2},^{de}Ω^{\frac{1}{2}} ) which leads to the following characterization of Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ). 3.7 Proposition. A continuous operator a : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) belongs to Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) if and only if it extends to a continuous operator a : C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) .

3.8 Corollary. For a, c ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) and a continuous operator b : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) we have abc ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ), i.e. Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) is a semi-ideal.

The following results on the asymptotic completeness of the double-edge calculus can be proved exactly as the asymptotic completeness in the classical case. 3.9 Proposition. Let a_{j} ∈ Ψ_{de}^{m−j,k}(X,^{de}Ω^{\frac{1}{2}} ), b_{j} ∈ Ψ_{de}^{m,k−j}(X,^{de}Ω^{\frac{1}{2}} ), and c_{j} ∈ Ψ_{de}^{m−j,k−j}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Then there exist a, b, c ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) such that a − ∑_{j=0}^{ℓ−1} a_{j} ∈ Ψ_{de}^{m−ℓ,k}(X,^{de}Ω^{\frac{1}{2}} ), b − ∑_{j=0}^{ℓ−1} b_{j} ∈ Ψ_{de}^{m,k−ℓ}(X,^{de}Ω^{\frac{1}{2}} ) and c − ∑_{j=0}^{ℓ−1} c_{j} ∈ Ψ_{de}^{m−ℓ,k−ℓ}(X,^{de}Ω^{\frac{1}{2}} ). The operators a, b, and c are uniquely determined up to Ψ_{de}^{−∞,k}(X,^{de}Ω^{\frac{1}{2}} ), Ψ_{de}^{m,−∞}(X,^{de}Ω^{\frac{1}{2}} ), and Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ), respectively. In that case, we write a ∼ ∑_{j=0}^{∞} a_{j}, b ∼ ∑_{j=0}^{∞} b_{j}, and c ∼ ∑_{j=0}^{∞} c_{j}.

Double-edge quantization and the complete double-edge symbol

Recall that a normal fibration of a closed p-submanifold P ⊆ Z of a manifold with corners Z is a diffeomorphism Θ : V ⟶^{≅} Θ(V) ⊆ Z where V ⊆ NP is an open neighborhood of P ⊆ NP, such that Θ|_{P} = id_{P}, and the induced map Θ_{∗} : NP ⟶ NP is the identity. Let now V ⊆ ^{de}T X resp. U ⊆ X_{de}^{2} be open neighbourhoods of X ∪ ^{de}T X|_{∂X} resp. Δ_{de} ∪ int ff^{de}. We call a diffeomorphism ^{de}Θ : V ⟶ U a double-edge quantization provided that up to the canonical diffeomorphism (12), ^{de}Θ is a normal fibration of the double-edge diagonal Δ_{de}, and ^{de}Θ has an extension to a diffeomorphism deΘ : V′ ⟶^{≅} U′ where V′ ⊆ X_{de}^{2} resp. U′ ⊆ ^{de}\overline{T} X are open sets satisfying V ∪ ^{de}\overline{T} X|_{∂X} ⊆ V′ resp. U ∪ ff^{de} ⊆ U′, and ^{de}Θ|_{^{de}\overline{T} X|_{∂X}} agrees with the diffeomorphism (14). Note that a double-edge quantization always exists. Indeed, choosing an embedding ^{de}T X ≅ NΔ_{de} ↪ T X_{de}^{2}|_{Δ_{de}} and integrating the (complete) vector fields lifted from ^{de}T X yields a normal fibration F_{1} of Δ_{de} after appropriate restrictions. On the other hand, the diffeomorphism (14) and the choice of a collar neighborhood of the boundary lead to a diffeomorphism F_{2} : ^{de}\overline{T} X|_{∂X×[0,ε)} ⟶^{≅} ff^{de} × [0, ε) that is a normal fibration near the double-edge diagonal Δ_{de}. Since two normal fibrations are homotopic through normal fibrations [28], we can patch the diffeomorphism F_{1} and F_{2} together to a double-edge quantization ^{de}Θ.

Furthermore, choose a function χ ∈ C_{c}^{∞}(\overline{U}) with χ = 1 in a neighborhood of ff^{de} ∪ Δ_{de}. By (42) we obtain a map

Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{−k}I_{cl,S}^{m}(^{de}T X, X; ϱ_{N}^{− \frac{2+2n+m}{2}} Ω^{\frac{1}{2}} (^{de}T X)) : a ⟼ ^{de}Θ^{∗}(χκ_{a}) ,

(43)

where the subscript S indicates that away from the zero-section X the distributions are Schwartz functions in the fibers of ^{de}T X, and where we have used the identification of the double-edge kernel density bundle ^{de}KD^{\frac{1}{2}} with the singular density bundle ϱ_{N}^{− \frac{2+2n+m}{2}} Ω^{\frac{1}{2}} (^{de}T X).

Fourier transform along the fibers maps the space on the right-hand side of (43) isomorphically onto

ϱ_{N}^{−k}S_{cl}^{m}(^{de}T^{∗}X, ϱ_{N}^{− \frac{2+2n+m}{2}} Ω^{\frac{1}{2}} (^{de}T^{∗}X)) ≅ ϱ_{N}^{−k}S_{cl}^{m}(^{de}T^{∗}X) ,  (44)

where in the last step we have used the canonical symplectic form ω_{de} to trivialize the density bundle. Finally, if ϱ_{σ} : ^{de}\overline{T}^{∗}X ⟶ \overline{ℝ}_{+} is a defining function for the double-edge cosphere bundle ^{de}S^{∗}X, then the space on the right hand-side of (44) is canonically isomorphic to ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X).

Note that the choice of a different cut-off function changes χκ_{a} by a kernel in Ċ^{∞}(X_{de}^{2},^{de}KD^{\frac{1}{2}} ), hence the corresponding element in ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X) by a function in Ċ^{∞}(^{de}\overline{T}^{∗}X); therefore, the map a ⟼ RC ∘ F ∘ ^{de}Θ^{∗}(χκ_{a}) induces a linear map

Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X)/Ċ^{∞}(^{de}\overline{T}^{∗}X)

that is onto, has kernel Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ), and is independent of the choice of the cut-off function χ but does depend on the choice of the double-edge quantization ^{de}Θ.

3.10 Definition. The isomorphism of vector spaces

deσ_{Θ} : Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} )/Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ \underbrace{ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X)/Ċ^{∞}(^{de}\overline{T}^{∗}X)}_{=:A^{m,k}}

is called the complete double-edge symbol with respect to the double-edge quantization ^{de}Θ.

Let us introduce some notations. For k, m ∈ ℝ let

π_{σ} : A^{m,k} ⟶ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X)/ϱ_{σ}^{∞}C^{∞}(^{de}\overline{T}^{∗}X) =: A_{σ}^{m,k}
π_{∂} : A^{m,k} ⟶ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X)/ϱ_{N}^{∞}C^{∞}(^{de}\overline{T}^{∗}X) =: A_{∂}^{m,k}
π_{∂,σ} : A^{m,k} ⟶ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X)/(ϱ_{σ}^{∞}C^{∞}(^{de}\overline{T}^{∗}X) + ϱ_{N}^{∞}C^{∞}(^{de}\overline{T}^{∗}X)) = A_{∂,σ}^{m,k}

be the projections. Note that the choice of a collar neighborhood near the boundaries fixes the following isomorphisms

A_{σ}^{m,k} ≅ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}S^{∗}X)[[ϱ_{σ}]] ≅ C^{∞}(^{de}S^{∗}X)[[ϱ_{σ}]] ,
A_{∂}^{m,k} ≅ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X|_{∂X})[[ϱ_{N}]] ≅ C^{∞}(^{de}\overline{T}^{∗}X|_{∂X})[[ϱ_{N}]] , and
A_{∂,σ}^{m,k} ≅ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}S^{∗}X|_{∂X})[[ϱ_{N}, ϱ_{σ}]] ≅ C^{∞}(^{de}S^{∗}X|_{∂X})[[ϱ_{N}, ϱ_{σ}]] ,

where, for a fixed ring R, R[[z]] denotes the ring of formal power series in z with coefficients in R. Moreover, for a ∈ A^{m,k} we have a = 0 if and only if π_{σ}(a) = 0 and π_{∂}(a) = 0.

By (41), the complete double-edge symbol ^{de}σ_{Θ} induces a bilinear map

∗ : A^{m_{1},k_{1}} × A^{m_{2},k_{2}} ⟶ A^{m_{1}+m_{2},k_{1}+k_{2}} : (a_{1}, a_{2}) ⟼ ^{de}σ_{Θ} (^{de}σ_{Θ}^{−1}(a_{1})(^{de}σ_{Θ}^{−1}(a_{2}))) (45)

depending only on the choice of the double-edge quantization ^{de}Θ. We are going to describe the properties of ∗. Let a_{i} ∈ A^{m_{i},k_{i}} be arbitrary. Then we have for φ, ψ ∈ C^{∞}(X) with ψ = 1 in a neighborhood of supp φ

φa_{1} ∗ (1 − ψ)a_{2} = (1 − ψ)a_{2} ∗ φa_{2} = 0 in A^{m_{1}+m_{2},k_{1}+k_{2}} ,  (46)

i.e. the product ∗ is local in the base variables. 3.11 Proposition. For each double-edge quantization, there exists a unique sequence (P_{j}^{Θ})_{j∈ℕ_{0}} of bidifferential operators on ^{de}T^{∗}X, whose coefficients are polynomials in the fiber variables of ^{de}T^{∗}X, such that

P_{j}^{Θ} (ϱ_{N}^{−k_{1}}S^{[m_{1}]}(^{de}T^{∗}X) × ϱ_{N}^{−k_{2}}S^{[m_{2}]}(^{de}T^{∗}X)) ⊆ ϱ_{N}^{(−k_{1}+k_{2})}S^{[m_{1}+m_{2}−j]}(^{de}T^{∗}X)

and P_{j}^{Θ} (ϱ_{N}^{−k_{1}}ϱ_{σ}^{−m_{1}}C^{∞}(^{de}\overline{T}^{∗}X) × ϱ_{N}^{−k_{2}}ϱ_{σ}^{−m_{2}}C^{∞}(^{de}\overline{T}^{∗}X))
⊆ ϱ_{N}^{−(k_{1}+k_{2})}ϱ_{σ}^{−(m_{1}+m_{2}−j)}C^{∞}(^{de}\overline{T}^{∗}X)  (48)

for all k_{i}, m_{i} ∈ ℝ; in particular P_{j}^{Θ} : A^{m_{1},k_{1}} × A^{m_{2},k_{2}} ⟶ A^{m_{1}+m_{2}−j,k_{1}+k_{2}} is well-defined. Moreover,

(a) P_{0}^{Θ}(f_{1}, f_{2}) = f_{1}f_{2},

(b) P_{1}^{Θ}(f_{1}, f_{2}) − P_{1}^{Θ}(f_{2}, f_{1}) = \frac{1}{i}\{f_{1}, f_{2}\}_{de} for f_{i} ∈ ϱ_{N}^{−k_{i}}ϱ_{σ}^{−m_{i}}C^{∞}(^{de}\overline{T}^{∗}X), and

(c) π_{σ}(a_{1} ∗ a_{2}) = ∑_{j=0}^{∞} π_{σ}(P_{j}^{Θ}(a_{1}, a_{2})) in A^{m_{1}+m_{2},k_{1}+k_{2}}.

Proof: Consider the following commutative diagram

deT X ⊇ V  ⟶^{^{de}Θ} U  ⊆ X_{de}^{2}
↓j^{de}  ↓j^{de}  ↓β_{de}^{2}  ↓β_{de}^{2}
T X ⊇ Ṽ  \underrightarrow{Θ̃} Ũ  ⊆ X^{2}

where Θ̃ : Ṽ := j^{de}(V) ⟶ Ũ := β_{de}^{2}(U) is uniquely determined by ^{de}Θ on Ṽ_{0} := Ṽ ∩ {ϱ_{N} > 0}, and is a diffeomorphism if restricted to Ṽ_{0}. Let 2X be the double of X, and, for ε > 0, choose a quantization

Θ_{ε} : T 2X ⊇ V′ ⟶^{≅} U′ ⊆ (2X)^{2}

with Ṽ ⊆ V′ and Θ_{ε}|_{Ṽ ∩{ϱ_{N} >ε}} = Θ̃|_{Ṽ ∩{ϱ_{N} >ε}}. Exactly as in (45), the quantization Θ_{ε} leads to a product ∗_{ε} on the spaces of complete symbols on 2X. Since 2X is a closed manifold, we know [17] that there exists a unique sequence of bidifferential operators (P_{j,ε}^{Θ})_{j∈ℕ_{0}} on T^{∗}2X with polynomial coefficients in the fiber variables such that P_{j,ε}^{Θ} is of total homogeneity −j in the fiber variables, and b_{1} ∗_{ε} b_{2} = ∑_{j=0}^{∞} P_{j,ε}^{Θ}(b_{1}, b_{2}). Moreover, P_{0,ε}(g_{1}, g_{2}) = g_{1}g_{2} and P_{1,ε}(g_{1}, g_{2}) − P_{1,ε}(g_{2}, g_{1}) = 1/i{g_{1}, g_{2}} is the Poisson bracket. Since the product ∗_{ε} is local, P_{j,ε}^{Θ}|_{{ϱ_{N} >ε_{0}}} is independent of ε < ε_{0} by the choice of Θ_{ε}; thus, P_{j} := lim_{ε→0} P_{j,ε}^{Θ}|_{{ϱ_{N} >0}} exists, and is a bidifferential operator on T^{∗}X|_{ϱ_{N} >0} of total homogeneity −j in the fiber variables, having smooth coefficients that are polynomials in the fiber variables. Let P_{j}^{Θ} be the pull-back of P_{j} under the canonical map (j^{de})^{∗} : ^{de}T^{∗}X|_{{ϱ_{N} >0}} ⟶ T^{∗}X|_{{ϱ_{N} >0}}. Then P_{j}^{Θ} satisfies (a), (b), and π_{σ}(a_{1} ∗ a_{2})|_{{ϱ_{N} >0}} = ∑ π_{σ}(P_{j}^{Θ}(a_{1}|_{{ϱ_{N} >0}}, a_{2}|_{{ϱ_{N} >0}})), and it remains to consider the behavior of P_{j}^{Θ} for ϱ_{N} → 0, i.e. to check (47), (48), and (c).

On the other hand, we already know π_{σ}(a_{1} ∗ a_{2}) ∈ A_{σ}^{m_{1}+m_{2},k_{1}+k_{2}}. Since elements in ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}S^{∗}X) are uniquely determined by their restriction to deS^{∗}X|_{{ϱ_{N} >0}}, we get (47), and (48) follows from (47). □

A similar result holds for the boundary part of ∗. For that, fix a collar neighborhood of the boundary ∂X. 3.12 Proposition. For each double-edge quantization ^{de}Θ, there exists a unique sequence (Q_{ℓ}^{Θ})_{ℓ∈ℕ_{0}} of bidifferential operators whose coefficients are polynomials in the fiber variables of ^{de}T^{∗}X|_{∂X} such that

Q_{ℓ}^{Θ} (ϱ_{N}^{−k_{1}}ϱ_{σ}^{−m_{1}}C^{∞}(^{de}\overline{T}^{∗}X|_{∂X}) × ϱ_{N}^{−k_{2}}ϱ_{σ}^{−m_{2}}C^{∞}(^{de}\overline{T}^{∗}X|_{∂X}))
⊆ ϱ_{N}^{−(k_{1}+k_{2}−ℓ)}ϱ_{σ}^{−(m_{1}+m_{2})}C^{∞}(^{de}\overline{T}^{∗}X|_{∂X}) and  (49)
Q_{ℓ}^{Θ} : A_{∂}^{m_{1},k_{1}} × A_{∂}^{m_{2},k_{2}} ⟶ A_{∂}^{m_{1}+m_{2},k_{1}+k_{2}−ℓ}  (50)

for all k_{i}, m_{i} ∈ ℝ. Moreover, we have

(a) Q_{0}^{Θ}(f_{1}, f_{2}) = f_{1}f_{2} for f_{i} ∈ ϱ_{N}^{−k_{i}}ϱ_{σ}^{−m_{i}}C^{∞}(^{de}T X|_{∂X}), and

(b) π_{∂}(a_{1} ∗ a_{2}) = ∑_{ℓ=0}^{∞} Q_{ℓ}^{Θ}(π_{∂}(a_{1}), π_{∂}(a_{2})) in A_{∂}^{m_{1}+m_{2},k_{1}+k_{2}−ℓ}.

Proof: Because of (46), we can use local product coordinates (x, y, z) near the boundary as in (1); let (x, y, z, S, U, Z) be the corresponding local coordinates on ^{de}T X. Suppose first that the double-edge quantization ^{de}Θ corresponds to the map ^{de}T X ∋ (x, y, z, S, U, Z) ⟼ (S, U, Z, x, y, z) ∈ X_{de}^{2} with respect to the coordinates (S, U, Z, x′, y′, z′) on X_{de}^{2} as in (7). Furthermore, let κ_{i} = κ̂_{i}(S, U, Z, x′, y′, z′) |dS dU dZ \frac{dx′}{x^{′2}} \frac{dy′}{x^{′2n}} \frac{dz′}{x^{′m}} |^{\frac{1}{2}} be the Schwartz kernel of a representative of (^{de}σ_{Θ}^{(m_{i},k_{i})})^{−1}(a_{i}) ∈ A^{m_{i},k_{i}}. Then a straightforward computation shows that the Schwartz kernel of the composition is

κ̂(S, U, Z, x, y, z) =
∫_{ℝ^{m}} ∫_{ℝ^{n}} ∫_{ℝ} κ̂_{1} (\frac{S − S′}{(1 + xS′)^{2}} , \frac{U − U′}{(1 + xS′)^{2}} , \frac{Z − Z′}{1 + xS′} , x + x^{2}S′, y + x^{2}U′, z + xZ′)
κ̂_{2}(S′, U′, Z′, x, y, z)\frac{dS′dU′dZ′}{(1 + xS′)^{2+2n+m}} .  (51)

A close inspection of (51) shows that Fourier transform in the (S, U, Z)-direc-tion and Taylor expansion with respect to x lead to a sequence of bidifferential operators (Q_{ℓ}^{Θ})_{ℓ∈ℕ_{0}} with the required properties. In particular, we have Q_{0}^{Θ}(f_{1}, f_{2}) = f_{1}f_{2}.

An arbitrary double-edge quantization ^{de}Θ is related to this special case essentially by a change of coordinates in ^{de}T X, thus, the corresponding operators have the same properties (see [16] or [17, Section 18.2]). □

Let us finally describe the relation between the sequences (P_{j}^{Θ})_{j∈ℕ_{0}} and (Q_{ℓ}^{Θ})_{ℓ∈ℕ_{0}}. By (48), P_{j}^{Θ} induces a map P_{j}^{Θ} : A_{∂}^{m_{1},k_{1}} ×A_{∂}^{m_{2},k_{2}} ⟶ A_{∂}^{m_{1}+m_{2}−j,k_{1}+k_{2}}. Let P_{j,ℓ}^{Θ} be the part of P_{j}^{Θ} that increases the degree in ϱ_{N} by ℓ ∈ ℕ_{0}, i.e. for f_{i} ∈ C^{∞}(^{de}\overline{T}^{∗}X|_{∂X})

P_{j}^{Θ}(ϱ_{N}^{−k_{1}}ϱ_{σ}^{−m_{1}}f_{1}, ϱ_{N}^{−k_{2}}ϱ_{σ}^{−m_{2}}f_{2}) = ∑_{ℓ=0}^{∞} \underbrace{P_{j,ℓ}^{Θ}(ϱ_{N}^{−k_{1}}ϱ_{σ}^{−m_{1}}f_{1}, ϱ_{N}^{−k_{2}}ϱ_{σ}^{−m_{2}}f_{2})}_{∈ϱ_{N}^{−(k_{1}+k_{2}−ℓ)}ϱ_{σ}^{−(m_{1}+m_{2}−j)}C^{∞}(^{de}\overline{T}^{∗}X|_{∂X})}

holds. Then (P_{j,ℓ}^{Θ})_{j,ℓ∈ℕ_{0}} satisfies P_{j,ℓ}^{Θ} : A_{∂}^{m_{1},k_{1}} × A_{∂}^{m_{2},k_{2}} ⟶ A_{∂}^{m_{1}+m_{2}−j,k_{1}+k_{2}−ℓ}, P_{j,ℓ}^{Θ} : A_{∂,σ}^{m_{1},k_{1}} × A_{∂,σ}^{m_{2},k_{2}} ⟶ A_{∂,σ}^{m_{1}+m_{2}−j,k_{1}+k_{2}−ℓ}, and

π_{∂,σ}(a_{1} ∗ a_{2}) = ∑_{j,ℓ∈ℕ_{0}} P_{j,ℓ}^{Θ}(π_{∂,σ}(a_{1}), π_{∂,σ}(a_{2})) in A_{∂,σ}^{m_{1}+m_{2},k_{1}+k_{2}} .  (52)

By (49), Q_{ℓ}^{Θ} induces a bilinear map Q_{ℓ}^{Θ} : A_{∂,σ}^{m_{1},k_{1}} × A_{∂,σ}^{m_{2},k_{2}} ⟶ A_{∂,σ}^{m_{1}+m_{2},k_{1}+k_{2}−ℓ}. Let us denote by Q_{ℓ,j}^{Θ}, j ∈ ℕ_{0}, the part of Q_{ℓ}^{Θ} that increases the degree in ϱ_{σ} by j, consequently, we obtain Q_{ℓ,j}^{Θ} : A_{∂}^{m_{1},k_{1}} × A_{∂}^{m_{2},k_{2}} ⟶ A_{∂}^{m_{1}+m_{2}−j,k_{1}+k_{2}−ℓ}, Q_{ℓ,j}^{Θ} : A_{∂,σ}^{m_{1},k_{1}} × A_{∂,σ}^{m_{2},k_{2}} ⟶ A_{∂,σ}^{m_{1}+m_{2}−j,k_{1}+k_{2}−ℓ}, and

π_{∂,σ}(a_{1} ∗ a_{2}) = ∑_{j,ℓ∈ℕ_{0}} Q_{ℓ,j}^{Θ}(π_{∂,σ}(a_{1}), π_{∂,σ}(a_{2})) in A_{∂,σ}^{m_{1}+m_{2},k_{1}+k_{2}} .  (53)

A comparison of (52) and (53) yields P_{j,ℓ}^{Θ} = Q_{ℓ,j}^{Θ} for all j, ℓ ∈ ℕ_{0}, hence

Q_{ℓ}^{Θ} := ∑_{j=0}^{∞} P_{j,ℓ}^{Θ} : A_{∂,σ}^{m_{1},k_{1}} × A_{∂,σ}^{m_{2},k_{2}} ⟶ A_{∂,σ}^{m_{1}+m_{2},k_{1}+k_{2}−ℓ} .

Since Q_{ℓ}^{Θ} is a bidifferential operator with polynomial coefficients in the fiber variables, this actually implies that for each ℓ there is only a finite number of P_{j,ℓ}^{Θ} with P_{j,ℓ}^{Θ} ≠ 0; hence, Q_{ℓ}^{Θ} = ∑_{j=0}^{∞} P_{j,ℓ}^{Θ} : A_{∂}^{m_{1},k_{1}} × A_{∂}^{m_{2},k_{2}} ⟶ A_{∂}^{m_{1}+m_{2},k_{1}+k_{2}−ℓ} is well-defined. In particular,

Q_{1}^{Θ}(f_{1}, f_{2}) − Q_{1}^{Θ}(f_{2}, f_{1}) = ∑_{j=0}^{∞} (P_{j,1}^{Θ}(f_{1}, f_{2}) − P_{j,1}^{Θ}(f_{2}, f_{1}))

is the part of {·, ·}_{de} that increases the degree in ϱ_{N} by 1. With respect to local coordinates (x, y, z, ξ, η, ζ) on ^{de}T^{∗}X this implies by (3)

Q_{1}^{Θ}(f_{1}, f_{2}) − Q_{1}^{Θ}(f_{2}, f_{1}) = \frac{1}{i}(x^{2}∂_{x}f_{1}∂_{ξ}f_{2} − x^{2}∂_{x}f_{2}∂_{ξ}f_{1} + x∂_{z}f_{1}∂_{ζ}f_{2}

− x∂_{z}f_{2}∂_{ζ}f_{1} + x(2η∂_{η}f_{1}∂_{ξ}f_{2} − 2η∂_{η}f_{2}∂_{ξ}f_{1}) + x(ζ∂_{ζ}f_{1}∂_{ξ}f_{2} − ζ∂_{ζ}f_{2}∂_{ξ}f_{1})) .

Similarly, we obtain an involution ♯ on A^{m,k} by

♯ : A^{m,k} ⟶ A^{m,k} : a ⟼ ^{de}σ_{Θ} (((^{de}σ_{Θ})^{−1}(a))^{♯}) ,  (54)

and exactly as for the ∗-product, we can describe the properties of a^{♯} using sequences of differential operators. For brevity, we only state the main result. 3.13 Theorem. For each double-edge quantization ^{de}Θ, there exists a sequences of differential operators (P_{j}^{Θ})_{j∈ℕ_{0}} on ^{de}T^{∗}X, (Q_{ℓ}^{Θ})_{ℓ∈ℕ_{0}} on ^{de}T^{∗}X|_{∂X}, and (P_{j,ℓ}^{Θ})_{j,ℓ∈ℕ_{0}} such that P_{j}^{Θ} increases the degree in ϱ_{σ} by j, Q_{ℓ}^{Θ} that in ϱ_{N} by ℓ, and P_{j,ℓ}^{Θ} that in ϱ_{σ} by j and that in ϱ_{N} by ℓ, and we have P_{0}^{Θ}(f) = f̄ for f ∈ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X), and Q_{0}^{Θ}(g) = ḡ for g ∈ ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X|_{∂X}). Moreover, for a ∈ A^{m,k} we have

π_{σ}(a^{♯}) = ∑_{j=0}^{∞} π_{σ}(P_{j}^{Θ}(a)) in A_{σ}^{m,k} ,
π_{∂}(a^{♯}) = ∑_{ℓ=0}^{∞} π_{∂}(Q_{ℓ}^{Θ}(a)) in A_{∂}^{m,k} , and
π_{∂,σ}(a^{♯}) = ∑_{j,ℓ=0}^{∞} π_{∂,σ}(P_{j,ℓ}^{Θ}(a)) in A_{∂,σ}^{m,k} .

For each ℓ ∈ ℕ_{0}, only a finite number of P_{j,ℓ}^{Θ} is different from 0, and we have Q_{ℓ}^{Θ} = ∑_{j=0}^{∞} P_{j,ℓ}^{Θ} : A_{∂}^{m,k} ⟶ A_{∂}^{m,k−ℓ}.

The joint double-edge symbol

Let C_{de}X := ∂(^{de}\overline{T}^{∗}X) be the boundary of the manifold with corners ^{de}\overline{T}^{∗}X. Then

C_{de}X = ^{de}S^{∗}X ∪^{de}\overline{T}^{∗}X|_{∂X} and ^{de}S^{∗}X ∩^{de}\overline{T}^{∗}X|_{∂X} = ^{de}S^{∗}X|_{∂X} .  (55)

Note that ^{de}S^{∗}X as well as ^{de}\overline{T}^{∗}X|_{∂X} are both smooth compact manifolds, with boundary ^{de}S^{∗}X|_{∂X}. Also C_{de}X is topologically a compact manifold without boundary, although it carries no natural smooth structure. 3.14 Definition. The space of all pairs (f_{σ}, f_{N}) ∈ C^{∞}(^{de}S^{∗}X)⊕C^{∞}(^{de}\overline{T}^{∗}X|_{∂X}) satisfying the compatibility condition

f_{σ}|_{^{de}S^{∗}X|_{∂X}} = f_{N}|_{^{de}S^{∗}X|_{∂X}}  (56)

is denoted by C^{∞}(C_{de}X). 3.15 Remark. Since C_{de}X is a compact topological space, we can consider the C^{∗}-algebra C(C_{de}X). Then

j_{de}^{∞} : C^{∞}(C_{de}X) ⟶ C(C_{de}X) : (f_{σ}, f_{N}) ⟼: ω ⟼ \{ \begin{matrix}f_{σ}(ω), & ω ∈ ^{de}S^{∗}X \\ f_{N}(ω), & ω ∈ ^{de}\overline{T}^{∗}X|_{∂X} \end{matrix}

is a well-defined injection because of (56), and the range j_{de}^{∞}(C^{∞}(C_{de}X)) of j_{de}^{∞} is a symmetric, dense subalgebra of C(C_{de}X) by the Stone-Weierstraß Theorem. For the sake of notational simplicity, we will often drop the map j_{de}^{∞}, and simply write f(ω) = (f_{σ}, f_{N})(ω) instead of j_{de}^{∞}(f)(ω).

Note that the choice of a boundary defining function ϱ_{N} : X ⟶ \overline{ℝ}_{+} and of a defining function ϱ_{σ} : ^{de}\overline{T}^{∗}X ⟶ \overline{ℝ}_{+} of the double-edge cosphere bundle ^{de}S^{∗}X fix a well-defined restriction map

R^{(m,k)}(·; ϱ_{σ}, ϱ_{N}) : ϱ_{N}^{−k}ϱ_{σ}^{−m}C^{∞}(^{de}\overline{T}^{∗}X) ⟶ C^{∞}(C_{de}X) .

3.16 Definition. The joint double-edge symbol

τ_{de}^{(m,k)} : (·; ϱ_{σ}, ϱ_{N}) : Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{∞}(C_{de}X)

is the composition of Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) → Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} )/Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) with R^{(m,k)}(·; ϱ_{σ}, ϱ_{N}) ∘^{de}σ_{Θ} : Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} )/Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{∞}(C_{de}X).

For a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ), we call the restrictions
deσ^{(m,k)}(a; ϱ_{σ}, ϱ_{N}) := τ_{de}^{(m,k)}(a; ϱ_{σ}, ϱ_{N})|_{^{de}S^{∗}X} ∈ C^{∞}(^{de}S^{∗}X) resp.
deσ_{N}^{(m,k)}(a; ϱ_{σ}, ϱ_{N}) := τ_{de}^{(m,k)}(a; ϱ_{σ}, ϱ_{N})|_{^{de}\overline{T}^{∗}X|_{∂X}} ∈ C^{∞}(^{de}\overline{T}^{∗}X|_{∂X})

the homogeneous principal resp. the normal symbol of a. 3.17 Lemma. The joint double-edge symbol does not depend on the choice of the quantization map ^{de}Θ. Proof: The homogeneous principal symbol^{de}σ^{(m,k)} is exactly the homogeneous principal symbol map for conormal distributions as discussed in [16] or [17, Theorem 18.2.11], hence it does not depend on the double-edge quantization deΘ. On the other hand, the restriction of the complete symbol to ^{de}\overline{T}^{∗}X|_{∂X} does not depend on ^{de}Θ by the definition of a double-edge quantization. □

3.18 Remark. Suppose ϱ′_{N} = fϱ_{N} with 0 < f ∈ C^{∞}(X) and ϱ′_{σ} = gϱ_{σ} with 0 < g ∈ C^{∞}(^{de}\overline{T}^{∗}X) are two equivalent defining functions. Then we have

τ_{de}^{(m,k)}(a; ϱ′_{σ}, ϱ′_{N}) = f^{k}g^{m} τ_{de}^{(m,k)}(a; ϱ_{σ}, ϱ_{N}) ,  (57)

thus, there is a smooth vector bundle E^{m,k} ⟶ C_{de}X such that the maps τ_{de}^{(m,k)}(a; ϱ_{σ}, ϱ_{N}) induce a well-defined section τ_{de}^{(m,k)}(a) ∈ C^{∞}(C_{de}X, E^{m,k}) independent of the choice of the defining functions ϱ_{σ} and ϱ_{N}. Here the space C^{∞}(C_{de}X, E^{m,k}) is defined analogous to Definition 3.14

For the sake of completeness, let us collect the main properties of the joint double-edge symbol. They are immediate consequences of Proposition 3.11, Proposition 3.12, and Theorem 3.13. 3.19 Proposition. The joint symbol τ_{de}^{(m,k)} : Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C^{∞}(C_{de}X) is a multiplicative ∗-homomorphism. Here the ∗-operation on the left-hand side is given by the formal adjoint ♯, and the multiplicativity has the obvious graded meaning. Moreover, the following short sequence

0 ⟶ Ψ_{de}^{m−1,k−1}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶^{τ_{de}^{(m,k)}} C^{∞}(C_{de}X) ⟶ 0 (58)

is exact, and τ_{de}^{(m,k)}(ϱ_{N}^{ℓ}aϱ_{N}^{−ℓ}; ϱ_{σ}, ϱ_{N}) = τ_{de}^{(m,k)}(a; ϱ_{σ}, ϱ_{N}) holds for ℓ ∈ ℝ and a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ).

Consequently, an operator whose joint symbol is invertible, is invertible modulo operators of lower order at the boundary and the lifted diagonal. A formal Neumann series together with the asymptotic completeness of Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) leads as usual to a better result. 3.20 Proposition. Let a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary with τ_{de}^{(m,k)}(a)(ω) ≠ 0 for all ω ∈ C_{de}X. Then there exists an operator b ∈ Ψ_{de}^{−m,−k}(X,^{de}Ω^{\frac{1}{2}} ) such that we have id−ab ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) and id−ba ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ). The operator b is uniquely determined up to Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ).

The proof of the following lemma reduces easily to a symbolic argument due to Hörmander [16, Proposition 2.2.2]. 3.21 Lemma. Let a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) be with τ_{de}^{(0,0)}(a) > 0 and a = a^{♯}. Then there exist b ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) and r ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) with b = b^{♯} and a = b^{2} + r.

General bundles

The double-edge calculus extends naturally to arbitrary vector bundles over X. Indeed, let E, F ⟶ X be two finite-dimensional complex vector bundles. Then we define Ψ_{de}^{m,k}(X; E, F) to be the space

Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) ⊗_{C^{∞}(X_{de}^{2})} C^{∞}(X_{de}^{2}, (β_{de}^{2})^{∗}(Hom (E ⊗^{de}Ω^{− \frac{1}{2}} , F ⊗^{de}Ω^{− \frac{1}{2}} )) . (59)

As one might expect, most of the properties for double-edge pseudodifferential operators remain true with the obvious modifications. For instance, a doubleedge pseudodifferential operator a ∈ Ψ_{de}^{m,k}(X; E, F) induces continuous linear operators a : Ċ^{∞}(X, E) ⟶ Ċ^{∞}(X, F) and a : C^{−∞}(X, E) ⟶ C^{−∞}(X, F). The joint double-edge symbol is then the map

τ_{de}^{(m,k)} : Ψ_{de}^{m,k}(X; E, F) ⟶ C^{∞}(C_{de}X; E, F)

where the space on the right hand side consists of all pairs

(f_{σ}, f_{N}) ∈ C^{∞}(^{de}S^{∗}X, \overline{p}^{∗}(Hom (E, F))) ⊕ C^{∞}(^{de}\overline{T}^{∗}X|_{∂X}, \overline{p}^{∗}(Hom (E, F)))

satisfying f_{σ}|_{^{de}S^{∗}X|_{∂X}} = f_{N}|_{^{de}S^{∗}X|_{∂X}}. Proposition 3.19 remains true.

4 Weighted double-edge Sobolev spaces

Let L^{2}(X,^{de}Ω^{\frac{1}{2}} ) be the Hilbert space completion of Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) with respect to the scalar product

< f, g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}= ∫_{X} f\overline{g} .

Boundedness of Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) on L^{2}-spaces

As in [16] the basic fact that double-edge operators in Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) act as bounded operators on L^{2}(X,^{de}Ω^{\frac{1}{2}} ) can be reduced to operators in the space Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) using Lemma 3.21. Since operators in Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) have Schwartz kernels in Ċ^{∞}(X^{2},^{de}Ω^{\frac{1}{2}} ) ⊆ L^{2}(X^{2},^{de}Ω^{\frac{1}{2}} ), they are Hilbert-Schmidt, in particular bounded on L^{2}(X,^{de}Ω^{\frac{1}{2}} ), and we have proved: 4.1 Theorem. Let a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Then a extends uniquely from Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) to a bounded operator a : L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ L^{2}(X,^{de}Ω^{\frac{1}{2}} ).

Later on we will also need a weighted version of L^{2}(X,^{de}Ω^{\frac{1}{2}} ). Let 𝔟 ∈ ℝ be arbitrary, and define

ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) := \{f ∈ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) : ϱ_{N}^{−𝔟}f ∈ L^{2}(X,^{de}Ω^{\frac{1}{2}} )\} .  (60)

The space ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) clearly becomes a Hilbert space with respect to the scalar product

< f, g >_{ϱN^{𝔟} L^{2}(X,^{de}Ω^{\frac{1}{2}} )}:=< ϱ_{N}^{−𝔟}f, ϱ_{N}^{−𝔟}g >_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} , f, g ∈ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) (61)

depending on the choice of the boundary defining function ϱ_{N}. Lemma 3.4 and Theorem 4.1 give 4.2 Corollary. Let a ∈ Ψ_{de}^{0,k}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Then a extends from

Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) to a bounded operator a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) .

Since Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) is dense in the Hilbert space ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) for any 𝔟 ∈ ℝ, the natural map Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) is one-to-one, and we can think of Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) as a subalgebra of the C^{∗}-algebra L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )). Note that besides the case 𝔟 = 0, the C^{∗}-structure on L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) depends on the choice of the boundary defining function ϱ_{N} : X ⟶ \overline{ℝ}_{+}.

4.3 Lemma. For any weight 𝔟 ∈ ℝ, the algebra Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is a symmetric subalgebra of the C^{∗}-algebra L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )). Proof: The Hilbert space adjoint of an operator a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is given by

a^{∗_{𝔟}} = ϱN^{𝔟}(ϱ_{N}^{−𝔟}aϱN^{𝔟})^{♯}ϱ_{N}^{−𝔟} ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) by Lemma 3.4. □

Weighted double-edge Sobolev spaces

This subsection follows the lines of an expanded version of [24], where b-Sobolev spaces on manifolds with corners were considered.

First, note that double-edge Sobolev spaces of integral order can easily be defined by induction.

H_{de}^{0}(X,^{de}Ω^{\frac{1}{2}} ) := L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ,
H_{de}^{m+1}(X,^{de}Ω^{\frac{1}{2}} ) := \{f ∈ H_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) : Diff_{de}^{1}(X,^{de}Ω^{\frac{1}{2}} )f ⊆ H_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} )\} .

(62)

for m ∈ ℕ_{0}. Since Diff_{de}^{1}(X,^{de}Ω^{\frac{1}{2}} ) is finitely generated as a C^{∞}(X)-module, we can replace it in (62) by a finite subset V ⊆ Diff_{de}^{1}(X,^{de}Ω^{\frac{1}{2}} ) without changing the spaces H_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ). Using the iterated graph norms with respect to the finite system V as in [14] resp. [21], we obtain a non-natural Hilbert space structure on the spaces H_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ). However, the structure of H_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) as so-called Hilbertable spaces does not depend on the choice of the set V, i.e. different choices lead to equivalent scalar products.

The next Proposition shows how to extend the definition of double-edge Sobolev spaces to a scale of Hilbert spaces ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ), 𝔟, s ∈ ℝ, by using the double-edge calculus. 4.4 Proposition. For each m ∈ ℕ_{0} and each 𝔟 ∈ ℝ we have

H_{de}^{m}(X,^{de}Ω^{\frac{1}{2}} ) =\{f ∈ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) : Ψ_{de}^{m,0}(X,^{de}Ω^{\frac{1}{2}} )f ⊆ L^{2}(X,^{de}Ω^{\frac{1}{2}} )\}(63)
ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) =\{f ∈ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) : Ψ_{de}^{0,𝔟}(X,^{de}Ω^{\frac{1}{2}} )f ⊆ L^{2}(X,^{de}Ω^{\frac{1}{2}} )\} (64)

Proof: For m ∈ 2ℕ_{0}, we use Theorem 4.1 and the existence of a parametrix b ∈ Ψ_{de}^{−m,0}(X,^{de}Ω^{\frac{1}{2}} ) for elliptic double-edge differential operators (Proposition 3.20). For m = 2ℓ + 1, we use the fact that each operator a ∈ Ψ_{de}^{m,0}(X,^{de}Ω^{\frac{1}{2}} ) has a decomposition of the form a = ∑_{j=1}^{M} p_{j}a_{j} + b, with p_{j} ∈ Diff_{de}^{1}(X,^{de}Ω^{\frac{1}{2}} ), a_{j} ∈ Ψ_{de}^{2ℓ,0}(X,^{de}Ω^{\frac{1}{2}} ), j = 1, …, M, and b ∈ Ψ_{de}^{2ℓ,0}(X,^{de}Ω^{\frac{1}{2}} ). □

4.5 Definition. For arbitrary 𝔟 ∈ ℝ and s ∈ ℝ, the space

ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) := \{f ∈ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) : Ψ_{de}^{s,𝔟}(X,^{de}Ω^{\frac{1}{2}} )f ⊆ L^{2}(X,^{de}Ω^{\frac{1}{2}} )\}

(65)

is called the double-edge Sobolev space of order s and weight 𝔟.

As an immediate consequence of this definition we obtain

ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ ϱ^{𝔟_{1}}H_{de}^{s_{1}}(X,^{de}Ω^{\frac{1}{2}} ) provided s − s_{1} ∈ ℕ_{0} and 𝔟 − 𝔟_{1} ∈ ℕ_{0} ,

(66)

and a (ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )) ⊆ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) for all a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) by

(41). With a little more effort one can actually prove a more precise characterization

ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ ϱ^{𝔟_{1}}H_{de}^{s_{1}}(X,^{de}Ω^{\frac{1}{2}} ) ⟺ s ≥ s_{1} and 𝔟 ≥ 𝔟_{1} .  (67)

We are now going to introduce on ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) the structure of a Hilbertable topological vector space. For each pair (s, 𝔟) ∈ ℝ^{2} fix an operator F_{(s,𝔟)} ∈ Ψ_{de}^{s,𝔟}(X,^{de}Ω^{\frac{1}{2}} ) such that

F_{(0,𝔟)} = ϱ_{N}^{−𝔟}id ∈ Ψ_{de}^{0,𝔟}(X,^{de}Ω^{\frac{1}{2}} ) , and  (68)
R_{(s,𝔟)} := id − F_{(−s,−𝔟)}F_{(s,𝔟)} ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) .  (69)

This is possible because of Proposition 3.20; we can now define a scalar product

on ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) by

< f, g >_{ϱN^{𝔟} H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )}:=< F_{(s,𝔟)}f, F_{(s,𝔟)}g >_{L^{2}} + < R_{(s,𝔟)}f, R_{(s,𝔟)}g >_{ϱN^{𝔟} L^{2}} (70)

for f, g ∈ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ). Let ∥f∥_{ϱN^{𝔟} H_{de}^{s}}^{2} = ∥F_{(s,𝔟)}f∥_{L^{2}}^{2} + ∥R_{(s,𝔟)}f∥_{ϱN^{𝔟} L^{2}}^{2} be the

corresponding norm. Note that the scalar product as well as the corresponding norm depend on the choice of the family of operators F_{(s,𝔟)} but we do not indicate this in the notation. Anyway, at least the structure of ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) as a Banachable topological vector space will be shown to be independent of the choice of F_{(s,𝔟)}. 4.6 Lemma. For all 𝔟, k, s, s′ ∈ ℝ and all a ∈ Ψ_{de}^{−∞,k}(X,^{de}Ω^{\frac{1}{2}} ) the map

a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s′}(X,^{de}Ω^{\frac{1}{2}} )

is bounded. Proof: Using (41), (69), the definition of ∥ · ∥_{ϱN^{𝔟} H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )}, and Corollary 4.2 we obtain

∥af∥_{ϱ_{N}^{𝔟−k}H_{de}^{s′}(X,^{de}Ω^{\frac{1}{2}} )}^{2}
≤ 2(∥ \underbrace{F_{(s′,𝔟−k)}aF_{(−s,−𝔟)}}_{∈Ψ_{de}^{−∞,0}(X,^{de}Ω^{\frac{1}{2}} )} F_{(s,𝔟)}f∥_{L^{2}}^{2} + ∥ \underbrace{F_{(s′,𝔟−k)}aR_{(s,𝔟)}}_{∈Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} )} f∥_{L^{2}}^{2}

+∥ \underbrace{R_{(s′,𝔟−k)}aF_{(−s,−𝔟)}}_{∈Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} )} F_{(s,𝔟)}f∥_{ϱ_{N}^{𝔟−k}L^{2}}^{2} + ∥ \underbrace{R_{(s′,𝔟−k)}aR_{(s,𝔟)}}_{∈Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} )} f∥_{ϱ_{N}^{𝔟−k}L^{2}}^{2})

≤ C (∥F_{(s,𝔟)}f∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}^{2} + ∥R_{(s,𝔟)}f∥_{ϱN^{𝔟} L^{2}(X,^{de}Ω^{\frac{1}{2}} )}^{2})
= C∥f∥_{ϱN^{𝔟} H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )}^{2} ,

which completes the proof. □

4.7 Theorem. Let a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Then a induces for all 𝔟, s ∈ ℝ a bounded operator

a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) .  (71)

Proof: Using (41), (69), Corollary 4.2, and Lemma 4.6 we obtain

∥af∥_{ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} )}^{2} ≤ 2(∥ \underbrace{F_{(s−m,𝔟−k)}aF_{(−s,−𝔟)}}_{∈Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} )} F_{(s,𝔟)}f∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}^{2} +

+ ∥ \underbrace{F_{(s−m,𝔟−k)}aR_{(s,𝔟)}}_{∈Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} )} f∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )}^{2} + ∥ \underbrace{R_{(s−m,𝔟−k)}a}_{∈Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} )} f∥_{ϱ_{N}^{𝔟−k}L^{2}(X,^{de}Ω^{\frac{1}{2}} )}^{2})

≤ C∥f∥_{ϱN^{𝔟} H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )}^{2}

which completes the proof. □

4.8 Corollary. Let 𝔟, s ∈ ℝ be arbitrary. Then a sequence (f_{j})_{j∈ℕ} in ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) converges to f ∈ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) if and only if the sequence (af_{j})_{j∈ℕ} converges for all a ∈ Ψ_{de}^{s,𝔟}(X,^{de}Ω^{\frac{1}{2}} ) in L^{2}(X,^{de}Ω^{\frac{1}{2}} ) to af. In particular, any two norms on ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) given by any two different families F_{(s,𝔟)} are equivalent.

4.9 Corollary. For all 𝔟, s ∈ ℝ, Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) is dense in ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ).

4.10 Corollary. For all 𝔟, s ∈ ℝ, the space ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) is complete, i.e. a Hilbertable topological vector space.

Note that the filtration of weighted double-edge Sobolev spaces satisfies:

Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) = ⋂_{𝔟,s∈ℝ} ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) and  (72)
C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) = ⋃_{𝔟,s∈ℝ} ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) .  (73)

The next result, though standard, is essential for the Fredholm theory in Section 5. 4.11 Proposition. If s > s_{1} and 𝔟 > 𝔟_{1} then the canonical embedding

ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ↪ ϱ^{𝔟_{1}}H_{de}^{s_{1}}(X,^{de}Ω^{\frac{1}{2}} ) is compact.

4.12 Corollary. For any a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) with τ_{de}^{(m,k)}(a) = 0 and any

𝔟, s ∈ ℝ, the bounded operator a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) is

compact.

General bundles

The results of this section extend directly to sections with values in general, finite-dimensional complex vector bundles over X. However, for the definition of the scalar product in the space H_{de}^{0}(X, E) = L_{de}^{2}(X, E) we have to fix a hermitian metric on E and a double-edge density on X. Alternatively, we could consider the bundle E ⊗^{de}Ω^{\frac{1}{2}} .

5 Fredholm theory for double-edge pseudod-

ifferential operators

Symbol reproducing families

As in the case of classical pseudodifferential operators on closed manifolds, the homogeneous principal symbol, as well as the normal symbol of a double-edge pseudodifferential operator a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) can be recovered from the action of a on a certain system of functions. The construction of such symbol reproducing families essentially goes back at least as far as [12]. Let us sketch the construction for the sake of completeness. For simplicity, we restrict ourselves to families reproducing the absolute value of the joint symbol. 5.1 Proposition. Let ω_{σ} ∈ ^{de}S^{∗}X be arbitrary. Then for each t > 0

there exists g_{t}^{ω_{σ}} ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) with ∥g_{t}^{ω_{σ}}∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = 1, g_{t}^{ω_{σ}} ⟶^{t→∞} 0 weakly

in L^{2}(X,^{de}Ω^{\frac{1}{2}} ), and lim_{t→∞} ∥ag_{t}^{ω_{σ}}∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = |^{de}σ^{(0,0)}(a)(ω_{σ})| for all doubleedge operators a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ). Proof: For \overline{p}(ω_{σ}) ∉ ∂X we use the standard construction for closed manifolds; so assume \overline{p}(ω_{σ}) ∈ ∂X. Let (x, y, z) : X ⊇ U ⟶ \overline{ℝ}_{+}×ℝ_{y}^{n}×ℝ_{z}^{m} be local product coordinates near \overline{p}(ω_{σ}), and let

(x, y, z, ξ, η, ζ) : ^{de}T^{∗}X ⊇ ^{de}T^{∗}X|_{U} ⟶ \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m} × ℝ_{ξ} × ℝ_{η}^{n} × ℝ_{ζ}^{m}

be the corresponding trivialization of the double-edge cotangent bundle ^{de}T^{∗}X near the boundary. Let now ω_{σ} = (0, 0, 0, ξ^{(0)}, η^{(0)}, ζ^{(0)}) ∈ ^{de}T^{∗}X \ {0} be a representative of ω_{σ} ∈ ^{de}S^{∗}X with respect to these coordinates. Choose v_{0} ∈ C_{c}^{∞}(ℝ_{+}) with ∫_{0}^{∞} |v_{0}(x)|^{2} \frac{dx}{x^{2+2n+m}} = 1, v_{1} ∈ C_{c}^{∞}(ℝ_{y}^{n}) with ∫_{ℝ_{y}^{n}} |v_{1}(y)|^{2}dy = 1, v_{2} ∈ C_{c}^{∞}(ℝ_{z}^{m}) with ∫_{ℝ_{z}^{m}} |v_{2}(z)|^{2}dz = 1, and define g_{t}^{ω_{σ}} ∈ C_{c}^{∞}(X,^{de}Ω^{\frac{1}{2}} ) in local coordinates up to the density factor |\frac{dx}{x^{2}} \frac{dy}{x^{2n}} \frac{dz}{x^{m}} |^{\frac{1}{2}} by

g_{t}^{ω_{σ}} : (x, y, z) ⟼ t^{− \frac{3+n+m}{2}} e^{−it^{7}xξ^{(0)}}e^{−it^{7}yη^{(0)}}e^{−it^{7}xzζ^{(0)}}v_{0}(t^{3}x)v_{1}(t^{5}y)v_{2}(t^{2}z) .

Using Lebesgue’s theorem twice, it is straightforward to check that g_{t}^{ω_{σ}} has the desired properties. □

5.2 Proposition. For any ω_{N} ∈ ^{de}\overline{T}^{∗}X|_{∂X} and any t > 0, there exists

a section g_{t}^{ω_{N}} ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) with ∥g_{t}^{ω_{N}} ∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = 1, g_{t}^{ω_{N}} ⟶^{t→∞} 0 weakly in

L^{2}(X,^{de}Ω^{\frac{1}{2}} ), and lim_{t→∞} ∥ag_{t}^{ω_{N}} ∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = |^{de}σ_{N}^{(0,0)}(a)(ω_{N})| for all doubleedge operators a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ). Proof: Let us start with the case ω_{N} ∈ ^{de}\overline{T}^{∗}X|_{∂X} \^{de}S^{∗}X|_{∂X}. Let (x, y, z) be local product coordinates near \overline{p}(ω_{N}) ∈ ∂X, and let

(x, y, z, ξ, η, ζ) : ^{de}T^{∗}X ⊇ ^{de}T^{∗}X|_{U} ⟶ \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m} × ℝ_{ξ} × ℝ_{η}^{n} × ℝ_{ζ}^{m}

be the corresponding trivialization of the double-edge cotangent bundle near the boundary. We can assume ω_{N} = (0, 0, 0, ξ^{(0)}, η^{(0)}, ζ^{(0)}) with respect to these coordinates. Choose v_{0} ∈ C_{c}^{∞}(ℝ_{+}) with ∫_{0}^{∞} |v_{0}(x)|^{2} \frac{dx}{x^{2+2n+m}} = 1, v_{1} ∈ C_{c}^{∞}(ℝ_{y}^{n}) with ∫_{ℝ_{y}^{n}} |v_{1}(y)|^{2}dy = 1, v_{2} ∈ C_{c}^{∞}(ℝ_{z}^{m}) with ∫_{ℝ_{z}^{m}} |v_{2}(z)|^{2}dz = 1, and define for t > 0 the section g_{t}^{ω_{N}} ∈ C_{c}^{∞}(X,^{de}Ω^{\frac{1}{2}} ) in local coordinates up to the density factor |\frac{dx}{x^{2}} \frac{dy}{x^{2n}} \frac{dz}{x^{m}} |^{\frac{1}{2}} by

g_{t}^{ω_{N}} (x, y, z) := t^{− \frac{2+n+m}{2}} e^{−iξ^{(0)}/x}e^{iyη^{(0)}/x^{2}}e^{izζ^{(0)}/x}v_{0}(t^{2}x)v_{1}(t^{3}y)v_{2}(tz) .

As in the proof of Proposition 5.1 a twofold application of Lebesgue’s theorem shows that g_{t}^{ω_{N}} has the desired properties for ω_{N} ∈ ^{de}\overline{T}^{∗}X|_{∂X} \^{de}S^{∗}X|_{∂X}. For ω_{N} ∈ ^{de}S^{∗}X|_{∂X} we can use the sequence from Proposition 5.1 because of the compatibility condition (56). □

5.3 Corollary. Let ω ∈ C_{de}X be arbitrary. Then for each t > 0 there

exists a g_{t}^{ω} ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) with ∥g_{t}^{ω}∥_{ϱN^{𝔟} L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = 1, g_{t}^{ω} ⟶^{t→∞} 0 weakly in

ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ), and lim_{t→∞} ∥ag_{t}^{ω}∥_{ϱN^{𝔟} L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = |τ_{de}^{(0,0)}(a)(ω)| for all double-

edge operators a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ). Proof: By Proposition 5.1 and Proposition 5.2 we find h_{t}^{ω} ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) with

∥h_{t}^{ω}∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = 1, h_{t}^{ω} ⟶^{t→∞} 0 weakly in L^{2}(X,^{de}Ω^{\frac{1}{2}} ), and

lim_{t→∞} ∥ah_{t}^{ω}∥_{L^{2}(X,^{de}Ω^{\frac{1}{2}} )} = |τ_{de}^{(0,0)}(a)|

for all a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ). Then g_{t}^{ω} := ϱN^{𝔟}h_{t}^{ω} ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) has all the desired properties by (61) and Proposition 3.19. □

Characterization of Fredholm operators in Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} )

Recall that by Corollary 4.2 any a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) induces a bounded operator

a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ).

5.4 Theorem. Let a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) and 𝔟 ∈ ℝ be arbitrary. Then the bounded operator a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) is Fredholm if and only if τ_{de}^{(0,0)}(a)(ω) ≠ 0 for all ω ∈ C_{de}X. Proof: The sufficiency is an immediate consequence of Proposition 3.20. Indeed, let b ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) be a parametrix as in Proposition 3.20. Then r_{0} := id − ab ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) and r_{1} = id − b_{0}a ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) extend by Corollary 4.12 to compact operators

r_{0} : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) resp.
r_{1} : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ,

and a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) is a Fredholm operator.

On the other hand, let a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) be a Fredholm operator. Then there exists an operator b ∈ L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) such that r := id − ba ∈ L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) is compact. Let ω ∈ C_{de}X be arbitrary, and suppose τ_{de}^{(0,0)}(a)(ω) = 0. Let g_{t}^{ω} ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) be as in Corollary 5.3. Since r is compact, we obtain

1 = ∥g_{t}^{ω}∥_{ϱN^{𝔟} L^{2}} ≤ ∥b∥_{L(ϱN^{𝔟} L^{2})}∥ag_{t}^{ω}∥_{ϱN^{𝔟} L^{2}} + ∥rg_{t}^{ω}∥_{ϱN^{𝔟} L^{2}} ⟶^{t→∞} 0 ,

i.e. a contradiction which completes the proof. □

By Theorem 5.4 the algebra Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is an algebra with a K(ϱN^{𝔟}L^{2})- symbolic structure in the C^{∗}-algebra L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) in the sense of [21, Section 2.4], thus, we obtain a characterization of the compact operators in Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) [21, Proposition 2.30]. 5.5 Corollary. Let 𝔟 ∈ ℝ be arbitrary. Then a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) extends to a compact operator a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) if and only if τ_{de}^{(0,0)}(a) = 0.

Characterization of Fredholm operators in Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} )

Exactly as in Theorem 5.4 we use Proposition 3.20 to get a sufficient condition for the Fredholm property of a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ). 5.6 Proposition. Let a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary with τ_{de}^{(m,k)}(a)(ω) ≠ 0

for all ω ∈ C_{de}X. Then a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) is a

Fredholm operator for all 𝔟, s ∈ ℝ.

As usual, we have elliptic regularity for double-edge pseudodifferential operators. 5.7 Corollary. Let a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) with τ_{de}^{(m,k)}(a)(ω) ≠ 0 for all ω ∈ C_{de}X and u ∈ C^{−∞}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary. Then we have

au ∈ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟹ u ∈ ϱ_{N}^{𝔟+k}H_{de}^{s+m}(X,^{de}Ω^{\frac{1}{2}} ) .

In particular, we have u ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) if au = 0. Proof: As in Proposition 3.20, let b ∈ Ψ_{de}^{−m,−k}(X,^{de}Ω^{\frac{1}{2}} ) be a double-edge operator with r_{0} := id − ba ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ). Then we have

u = b \underbrace{au}_{∈ϱN^{𝔟} H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )} +r_{0}u ∈ ϱ_{N}^{𝔟+k}H_{de}^{s+m}(X,^{de}Ω^{\frac{1}{2}} )

by Theorem 4.7. The identity Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) = ⋂_{𝔟,s∈ℝ} ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) completes the proof. □

5.8 Proposition. For arbitrary k, m ∈ ℝ there exists Λ_{m,k} ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) such that the operator Λ_{m,k} : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) extends to an isomorphism

Λ_{m,k}^{s,𝔟} = Λ_{m,k} : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} )

for any 𝔟, s ∈ ℝ, and τ_{de}^{(m,k)}(Λ_{m,k}; ϱ_{σ}, ϱ_{N})(ω) = 1 for all ω ∈ C_{de}X. Proof: By (58), there exists a ∈ Ψ_{de}^{\frac{m}{2} , \frac{k}{2}} (X,^{de}Ω^{\frac{1}{2}} ) with τ_{de}^{( \frac{m}{2} , \frac{k}{2} )}(a) = 1, thus, we find an operator b ∈ Ψ_{de}^{− \frac{m}{2} ,− \frac{k}{2}} (X,^{de}Ω^{\frac{1}{2}} ) with r := id−ba ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) by Proposition 3.20. Then the operator Λ_{m,k} := a^{♯}a+r^{♯}r ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) satisfies τ_{de}^{(m,k)}(Λ_{m,k}; ϱ_{σ}, ϱ_{N}) = 1 and Λ_{m,k} : Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) extends to

a Fredholm operator Λ_{m,k}^{s,𝔟} : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) for all

𝔟, s ∈ ℝ by Proposition 5.6.

Let f ∈ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) be with Λ_{m,k}^{s,𝔟}f = 0. By Corollary 5.7 we have f ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ), hence 0 =< Λ_{m,k}f, f >_{L^{2}}=< af, af >_{L^{2}} + < rf, rf >_{L^{2}}, i.e. af = rf = 0; thus, we get f = baf + rf = 0, and Λ_{m,k}^{s,𝔟} is one-to-one.

Similarly, the adjoint (Λ_{m,k}^{s,𝔟})^{∗} : ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} )

of Λ_{m,k}^{s,𝔟} is one-to-one, too. Since the range R(Λ_{m,k}^{s,𝔟}) is closed, the operator

Λ_{m,k}^{s,𝔟} : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) is an isomorphism. □

These so-called order reducing operators can be used to reduce the general case Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) to the case Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ).

5.9 Proposition. Let 𝔟, s ∈ ℝ be arbitrary. Then a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) induces a compact operator a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) if and only if τ_{de}^{(m,k)}(a) = 0.

5.10 Theorem. Let 𝔟, s ∈ ℝ be arbitrary. Then a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) extends to a Fredholm operator a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) if and only if τ_{de}^{(m,k)}(a)(ω) ≠ 0 for all ω ∈ C_{de}X.

5.11 Remark. Since the joint symbol τ_{de}^{(m,k)} does not depend on the parameters 𝔟, s ∈ ℝ, we see that an operator a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ) is a Fredholm

operator a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) for one pair (𝔟, s) ∈ ℝ^{2} if

and only if it is Fredholm for every pair (𝔟, s) ∈ ℝ^{2}. By Proposition 3.20 we can construct in that case a parametrix up to remainders in Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) within Ψ_{de}^{−m,−k}(X,^{de}Ω^{\frac{1}{2}} ).

General bundles

The previous results extend naturally to the case of operators between arbitrary, finite-dimensional complex vector bundles. Let us only state the main result. The following notion will be useful in the sequel. 5.12 Definition. An operator a ∈ Ψ_{de}^{m,k}(X; E, F) is said to be fully elliptic provided the linear maps τ_{de}^{(m,k)}(a)(ω) : E_{\overline{p}(ω)} ⟶ F_{\overline{p}(ω)} are invertible for all ω ∈ C_{de}X. 5.13 Theorem. Let 𝔟, s ∈ ℝ be arbitrary. Then a ∈ Ψ_{de}^{m,k}(X; E, F) extends to a Fredholm operator a : ϱN^{𝔟}H_{de}^{s}(X, E) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X, F) if and only if it is fully elliptic.

An index formula for double-edge operators

Let a ∈ Ψ_{de}^{m,k}(X; E, F) be fully elliptic. Then a induces a Fredholm operator a : ϱN^{𝔟}H_{de}^{s}(X, E) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X, F) by Theorem 5.13. We are going to compute the index of a in terms of its joint symbol τ_{de}^{(m,k)}(a). We follow more or less the approach in [32, Section 6.5] for the scattering calculus. Note that by means of order reducing operators as in Proposition 5.8 we can reduce the problem to a fully elliptic element a ∈ Ψ_{de}^{0,0}(X; E, F).

5.14 Proposition. Let a ∈ Ψ_{de}^{0,0}(X; E, F) be fully elliptic. Then there exists a continuous path

γ : [0, 1] ⟶ Ψ_{de}^{0,0}(X; E, F) : t ⟼ a_{t} ,

such that a_{t} is fully elliptic for each t ∈ [0, 1], a_{0} = a, and a_{1} is a bundleisomorphism in a collar neighbourhood of the boundary. Proof: Let us sketch the construction of the homotopy. We start with a homotopy in the invertible elements of the space C^{∞}(C_{de}X; E, F). First, by blowing up ∂X ↪ ^{de}\overline{T}^{∗}X|_{∂X} we obtain a homotopy to a symbol b_{1} that is constant along the fibres of \overline{p} : ^{de}\overline{T}^{∗}X|_{∂X} ⟶ ∂X in an open neighbourhood of the zero-section ∂X, formaly b_{1}(0, y, z, ξ, η, ζ) = b_{1}(0, y, z, 0, 0, 0) for (ξ, η, ζ) near 0.

Note that ^{de}\overline{T}^{∗}X|_{∂X} and ^{de}S^{∗}X are two compact manifolds with the same boundary ^{de}S^{∗}X|_{∂X}. Thus, choosing a collar neigbourhood of ^{de}S^{∗}X|_{∂X} in ^{de}\overline{T}^{∗}X|_{∂X} resp. ^{de}S^{∗}X we can glue ^{de}\overline{T}^{∗}X|_{∂X} and ^{de}S^{∗}X together to a smooth manifold, i.e. we obtain a non-natural smooth structure on C_{de}X. By integrating a radial vector field using this smooth structure across the corner we get a homotopy to a symbol b_{2} that is the symbol of a bundle isomorphism in a neighbourhood of ^{de}\overline{T}^{∗}X|_{∂X}, i.e. formally we have b_{2}(x, y, z, ω) = f(y, z) for all sufficiently small x and all ω ∈ C_{de}X with \overline{p}(ω) = (x, y, z), where \overline{p} : ^{de}\overline{T}^{∗}X ⟶ X is the canonical projection.

Using a continuous map op_{de}^{(0,0)} : C^{∞}(C_{de}X; E, F) ⟶ Ψ_{de}^{0,0}(X; E, F) we obtain a continuous path in the fully elliptic elements in Ψ_{de}^{0,0}(X; E, F) from a to an operator that is up to an element in Ψ_{de}^{−1,−1}(X; E, F) a bundle-isomorphism in a collar neighbourhood of the boundary. Using a convex homotopy to kill this term completes the proof. □

Fix now a collar neighbourhood of the boundary of X such that a_{1} restricted to it is a bundle isomorphism, and let 2X := X ∪_{∂X} X =: X_{0}∪_{∂X} X_{1} be the smooth compact manifold without boundary obtained by gluing along the boundary. The manifold 2X is said to be the double of X. Because of the special structure of a_{1} near the boundary, there exists an elliptic pseudodifferential operator ã ∈ Ψ^{0}(X; E, F) such that ãf = a_{1}f whenever f is supported in X_{0} or X_{1}. In particular, ã induces a Fredholm operator ã : L^{2}(X, E) ⟶ L^{2}(X, F). Note that we have ãf = 0 if and only if a_{1}(f|_{X_{0}}) = 0 and a_{1}(f|_{X_{1}}) = 0 because ã is simply the bundle isomorphism a_{1} near ∂X and coincides with a_{1} in X_{0} resp. X_{1}. Thus, we get the following lemma. 5.15 Lemma. ind ã = 2 ind a_{1} = 2 ind a.

Let us briefly recall the definition of the topological index map for closed manifolds following [23]. The homogeneous principal symbol of an elliptic ele-

ment b ∈ Ψ^{0}(2X; E, F) induces an isomorphism π^{∗}E|_{𝕊(T^{∗}2X)} ⟶^{σ^{(0)}(b)} π^{∗}F |_{𝕊(T^{∗}2X)},

and hence an element [σ^{(0)}(b)] ∈ K(\overline{T}^{∗}2X, 𝕊(T^{∗}2X)) where \overline{T}^{∗}2X ⟶ 2X denotes the radial compactification of the cotangent bundle T^{∗}2X with boundary ∂(\overline{T}^{∗}(2X)) = 𝕊(T^{∗}2X), the cosphere bundle of 2X (cf. [23, Definition 1.9.23, Proposition 1.9.25]).

The choice of a Riemannian metric on T 2X leads to a diffeomorphism (\overline{T}^{∗}2X, 𝕊(T^{∗}2X)) ≅ (𝔻2X, 𝕊2X) where 𝔻2X resp. 𝕊2X is the disc resp. the sphere bundle of the tangent bundle T 2X ⟶ 2X.

Finally, let f : 2X ⟶ ℝ^{N} be an imbedding where N ∈ ℕ is a large integer. Then the shriek construction gives a map, the so-called topological index

ind^{t} := q_{!}f_{!} : K(𝔻2X, 𝕊2X) ⟶ ℤ ,

where q : T ℝ^{N} ⟶ {pt} is the canonical scrunch map [23, Definition 3.13.1], and the Atiyah-Singer index theorem (see [1] or [23, Theorem 3.13.2]) for ã gives

ind ã = ind^{t}[σ^{(0)}(ã)] .  (74)

We are now going to express the right-hand side of (74) in terms depending only on the joint symbol τ_{de}^{(0,0)}(a). First, note that by ellipticity the joint

symbol τ_{de}^{(0,0)}(a) induces an isomorphism \overline{p}^{∗}E|_{C_{de}X} ⟶^{τ_{de}^{(0,0)}(a)} \overline{p}^{∗}F |_{C_{de}X}, and hence

an element [τ_{de}^{(0,0)}(a)] ∈ K(^{de}\overline{T}^{∗}X, C_{de}X). Since a and a_{1} are homotopic through fully elliptic operators we certainly have [τ_{de}^{(0,0)}(a)] = [τ_{de}^{(0,0)}(a_{1})].

Note that there is a non-canonical isomorphism ^{de}T^{∗}X ⟶ T X of vector bundles over X. However, the corresponding isomorphism in K-theory K(^{de}\overline{T}^{∗}X, C_{de}X) ⟶ K(\overline{T} X, ∂\overline{T} X) does not depend on the choice of this isomorphism or on the choice of the Riemannian metric because any two choices are homotopic, hence not seen by K-Theory.

Consider the map Φ : K(\overline{T} X, ∂\overline{T} X) ⊕ K(\overline{T} X, ∂\overline{T} X) ⟶ K(𝔻2X, 𝕊2X) given by:

K(\overline{T} X, ∂\overline{T} X) ⊕ K(\overline{T} X, ∂\overline{T} X) ⟶^{≅} K̃(\overline{T} X/∂\overline{T} X) ⊕ K̃(\overline{T} X/∂\overline{T} X) (75)
⟶^{≅} K̃(\overline{T} X/∂\overline{T} X ∨ \overline{T} X/∂\overline{T} X)  (76)
⟶^{≅} K̃(𝔻2X/(𝕊2X ∪ 𝔻2X|_{∂X}))  (77)
⟶ K̃(𝔻2X/𝕊2X)  (78)
⟶^{≅} K(𝔻2X, 𝕊2X)  (79)

where (75) resp. (79) are simply the definition of the relative K-Theory [23, Definition 1.9.12], (76) is the wedge-isomorphism in a reduced cohomology theory [43, Proposition 7.36], (77) is induced by the homeomorphism of pointed spaces \overline{T} X/∂\overline{T} X ∨ \overline{T} X/∂\overline{T} X ≈ DZX/(𝕊2X ∪ 𝔻2X|_{∂X}) with ∂X = X_{0} ∩ X_{1}, and, finally, (78) is induced by the map 𝔻2X/𝕊2X ⟶ 𝔻2X/(𝕊2X ∪ 𝔻2X|_{∂X}).

As an immediate consequence of the definition of the map Φ and that of the K-Theory classes [τ_{de}^{(0,0)}(a)] and [σ^{(0)}(ã)], we obtain: 5.16 Proposition. [σ^{(0)}(ã)] = Φ([τ_{de}^{(0,0)}(a)] ⊕ [τ_{de}^{(0,0)}(a)]).

Let α : 2X ⟶ 2X be the flip, i.e. the diffeomorphism that interchanges X_{0} and X_{1} in 2X. Then we obtain by the naturality of the induced maps α^{∗}Φ([τ_{de}^{(0,0)}(a)] ⊕ 0) = Φ(0 ⊕ [τ_{de}^{(0,0)}(a)]). On the other hand, since fα as well as f are embeddings of 2X into ℝ^{N}, and the topological index does not depend on the embedding, we have ind^{t} ∘ α^{∗} = ind^{t}. Therefore, defining

Φ_{0} : K(^{de}\overline{T}^{∗}X, C_{de}X) ⟶ K(𝔻2X, 𝕊2X) : τ ⟼ Φ(τ ⊕ 0)

we get 2 ind^{t}Φ_{0}([τ_{de}^{(0,0)}(a)]) = ind^{t}Φ([τ_{de}^{(0,0)}(a)] ⊕ τ_{de}^{(0,0)}(a)]), and we have finally proved our index theorem. 5.17 Theorem. Let a ∈ Ψ_{de}^{0,0}(X; E, F) be fully elliptic. Then the operator a : L^{2}(X, E) ⟶ L^{2}(X, F) is a Fredholm operator with index

ind a = ind^{t}Φ_{0}([τ_{de}^{(0,0)}(a)]) .

A double-edge operator of index 1

The proof of the following theorem was suggested by R. B. Melrose.

5.18 Theorem. There exists L ∈ ℕ and a fully elliptic double-edge operator a ∈ Ψ_{de}^{0,0}(X; ℂ^{L} ⊗^{de}Ω^{\frac{1}{2}} , ℂ^{L} ⊗^{de}Ω^{\frac{1}{2}} ) = M_{L}(Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} )) such that a extends to a Fredholm operator a : L^{2}(X, ℂ^{L} ⊗^{de}Ω^{\frac{1}{2}} ) ⟶ L^{2}(X, ℂ^{L} ⊗^{de}Ω^{\frac{1}{2}} ) with index ind a = 1 Proof: Let N := dim X. Since the unit sphere S^{2N−1} ⊆ ℝ_{t}^{N} × ℝ_{ϑ}^{N} is odd dimensional, there exists L ∈ ℕ and a smooth map f_{0} : S^{2N−1} ⟶ GL(L; ℂ) such that deg f_{0} = 1 [3], i.e. we have

deg f_{0} = \frac{(N − 1)!}{(2N − 1)!(2πi)^{N}} ∫_{S^{2N−1}} tr(f_{0}^{−1}df_{0})^{2N−1} = 1 ,

where tr is the usual trace of matrices. By the homotopy invariance of the degree and eventually by multiplying with an element in GL(L; ℂ) we find a smooth function f_{1} : S^{2N−1} ⟶ GL(L; ℂ) of degree deg f_{1} = 1, such that f_{1}(t, ϑ) = id_{ℂ^{L}} for all (t, ϑ) ∈ S^{2N−1} with |t| ≥ \frac{1}{4}. Note that there is a homeomorphism

\{(t, ϑ) ∈ S^{2N−1} : |t| ≤ \frac{1}{4}\} ⟶^{≈} \{|t| ≤ \frac{1}{4}\} × S_{ϑ}^{N−1} .  (80)

Using an appropriate coordinate patch int X ⊇ U ⟶ ℝ_{t}^{N}, and the corresponding trivialization of the double-edge cotangent bundle ^{de}T^{∗}X|_{U} ⟶ ℝ_{t}^{N} × ℝ_{ϑ}^{N} and the double-edge cosphere bundle

deS^{∗}X|_{U} ⟶ ℝ_{t}^{N} × S_{ϑ}^{N−1} ,  (81)

we obtain a map f_{2} : C_{de}X ⟶ GL(L; ℂ) that coincides with f_{1} in ^{de}S^{∗}X|_{|t|≤ \frac{1}{4}} up to the obvious identifications given by (80) and (81), and is up to some density factor equal to the identity id_{ℂ^{L}} for |t| ≥ \frac{1}{4}.

Let a ∈ Ψ_{de}^{0,0}(X; ℂ^{L} ⊗ ^{de}Ω^{\frac{1}{2}} , ℂ^{L} ⊗ ^{de}Ω^{\frac{1}{2}} ) be with τ_{de}^{(0,0)}(a) = f_{2}. Without loss of generality we can assume that a is the identity outside the coordinate patch U – otherwise, we use a convex homotopy to kill the term in Ψ_{de}^{−1,−1}(X; ℂ^{L} ⊗ ^{de}Ω^{\frac{1}{2}} , ℂ^{L} ⊗ ^{de}Ω^{\frac{1}{2}} ). Thus, we can apply the same doubling procedure as explained before Lemma 5.15 and obtain an elliptic operator ã ∈ Ψ^{0}(2X; ℂ^{L} ⊗ Ω^{\frac{1}{2}} , ℂ^{L} ⊗ Ω^{\frac{1}{2}} ) with ind ã = 2 ind a.

On the other hand, up to this localized part in |t| ≤ \frac{1}{4} the pseudodifferential operator ã is simply the identity on the closed manifold 2X, and we can use the reasoning of [10] to reduce the computation of ind ã to a computation in local coordinates in ℝ_{t}^{N}, i.e. to Fedosov’s index formula which finally gives

ind a = \frac{1}{2} ind ã = degf_{1} = deg f_{0} = 1 ,

and completes the proof. □

5.19 Remark. A close inspection of the proof shows that this construction works for all algebras of pseudodifferential operators that coincide with the usual pseudodifferential calculus on the regular part of the manifold.

6 C^{∗}- and Ψ^{∗}- algebras of double-edge opera-

tors of order 0

Of particular interest is the space Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) because it is not only a vector space but also an algebra by (41), thus, we can use algebraic tools to investigate Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) and related algebras.

C^{∗}-algebras and symbolic structure

Let 𝔟 ∈ ℝ be arbitrary. Then Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) can be realized as a symmetric subalgebra of the C^{∗}-algebra L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) by Lemma 4.3. However, note that besides the case 𝔟 = 0 of no weight, the Hilbert space structure, and hence also the C^{∗}-algebra structure, depend on the choice of the boundary defining function ϱ_{N}. 6.1 Lemma. Let 0 < k, m ∈ ℝ ∪ {∞} be arbitrary. Then we have for all

𝔟 ∈ ℝ

\overline{Ψ_{de}^{−m,−k}(X,^{de}Ω^{\frac{1}{2}} )}^{L(ϱN^{𝔟} L^{2}(X,^{de}Ω^{\frac{1}{2}} ))} = K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) .  (82)

Proof: Because of Corollary 4.12, it remains to prove that K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) is contained in the left-hand side of (82). Since Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) is dense in ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) it suffices to check that for any two f, g ∈ Ċ^{∞}(X,^{de}Ω^{\frac{1}{2}} ) the operator

a := f ⊗ g : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) : φ ⟼< φ, g >_{ϱN^{𝔟} L^{2}(X,^{de}Ω^{\frac{1}{2}} )} f

belongs to Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ). Since the Schwartz kernel k_{a} of a = f ⊗ g is given by k_{a} = f ⊗ϱ_{N}^{−2𝔟}ḡ ∈ Ċ^{∞}(X^{2},^{de}Ω^{\frac{1}{2}} ), we have a ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) which completes the proof. □

By Theorem 5.4, Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is an algebra with K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ))- symbolic structure in the C^{∗}-algebra L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) in the sense of [21, Section 2.4.3]. The corresponding symbol map is given by the joint double-edge symbol τ_{de}^{(0,0)} : Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C(C_{de}X). Let us denote by B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) the closure of Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) in the C^{∗}-algebra L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )). 6.2 Proposition. The joint symbol τ_{de}^{(0,0)} : Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C(C_{de}X) extends for each 𝔟 ∈ ℝ to a morphism τ_{de}^{(𝔟)} : B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ C(C_{de}X) of C^{∗}-algebras, and induces the following short exact sequence

0 ⟶ K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) ⟶ B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) ⟶^{τ_{de}^{(𝔟)}} C(C_{de}X) ⟶ 0 .  (83)

Proof: Since Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is an algebra with symbolic structure, it suffices to

note that C^{∞}(C_{de}X) is dense in C(C_{de}X), K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) ⊆ B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )

by Lemma 6.1, and to apply [21, Proposition 2.31]. □

Since the C^{∗}-algebra B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) is essentially commutative, i.e. commutative modulo the ideal of compact operators, the description of the set of irreducible representations and the Jacobson topology on it are immediate, namely define for ω ∈ C_{de}X

π_{ω} : B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ℂ : a ⟼ τ_{de}^{(𝔟)}(a)(ω) ,

and denote by B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )^{}the spectrum of the C^{∗}-algebra B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ), i.e. the set of all equivalence classes of non-zero irreducible representations of B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ). 6.3 Corollary. The representations π_{ω} : B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ℂ, ω ∈ C_{de}X, are irreducible and pairwise inequivalent. Moreover, an arbitrary, non-zero irreducible representation of B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) is unitary equivalent to either the identity representation id : B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) or to one π_{ω} for some ω ∈ C_{de}X. Finally, the corresponding canonical map

Φ : T := C_{de}X⊎]\{[id]\} ⟶ B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )^{}

is a homeomorphism provided B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )^{}is endowed with the Jacobson topology and T with the topology: T (T) = {∅}∪{{[id]}⊎V : V ⊆ C_{de}X open }.

Proof: For the description of the Jacobson topology see for instance [20, Proposition 5.1] or [36]. The rest is clear. □

Ψ^{∗}-algebras of double-edge operators

Let us first show the crucial property that a double-edge pseudodifferential operator a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is invertible in the algebra Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) if and only if it is invertible in the C^{∗}-algebra of all bounded operators on the Hilbert space ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) for one weight 𝔟 ∈ ℝ. This property is also known as spectral invariance. Recall that a symmetric, spectrally invariant, continuously embedded Fréchet subalgebra of a C^{∗}-algebra is called a Ψ^{∗}-algebra [13, Definition 5.1]. 6.4 Theorem. For any 𝔟 ∈ ℝ, Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) ↪ L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) is a Ψ^{∗}-algebra. Proof: Since there is an obvious Fréchet topology on Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) that is stronger than the norm topology of L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )), and Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is symmetric with respect to the ∗-operation of t L(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} )) by Lemma 4.3, it remains to check the spectral invariance. Suppose that a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is invertible as a bounded operator a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ). By Theorem 5.4 we have τ_{de}^{(0,0)}(a)(ω) ≠ 0 for all ω ∈ C_{de}X, hence there exists a double-edge operator b ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) with r_{0} := id − ba ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) and r_{1} := id − ab ∈ Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) by Proposition 3.20. Thus, we obtain

a^{−1} = b + a^{−1}r_{1} = b + br_{1} + r_{0}a^{−1}r_{1} ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} )

by the semi-ideal property of the space Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) (cf. Corollary 3.8). This completes the proof. □

By [13, Bemerkung 5.7] also the generalized inverses of Fredholm operators in a Ψ^{∗}-algebra belong to the Ψ^{∗}-algebra. Recall that the generalized inverse of a Fredholm operator a is the uniquely determined operator inverting a up to the orthogonal projection onto kernel and cokernel, respectively.

6.5 Corollary. Let a ∈ Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) be arbitrary with τ_{de}^{(0,0)}(a)(ω) ≠ 0 for all ω ∈ C_{de}X. Then for all 𝔟 ∈ ℝ the generalized inverse g^{(𝔟)} of the Fredholm

operator a : ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ) belongs to Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ).

Note that for 𝔟 ≠ 0 the generalized inverse depends on the Hilbert space structure of the space ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ), i.e. on the choice of the boundary defining function ϱ_{N}.

Exactly as the spectral invariance in Theorem 6.4 we prove the following fact. 6.6 Proposition. Let 𝔟, s ∈ ℝ be arbitrary, and let a ∈ Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} )

be such that a : ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) is invertible as a

bounded operator for some 𝔟, s ∈ ℝ. Then there exists an operator b ∈ Ψ_{de}^{−m,−k}(X,^{de}Ω^{\frac{1}{2}} )

such that we have a^{−1} = b : ϱ_{N}^{𝔟−k}H_{de}^{s−m}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ϱN^{𝔟}H_{de}^{s}(X,^{de}Ω^{\frac{1}{2}} ).

K-Theory of the algebras B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) and Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} )

With the previous results the computation of the K-theory of the C^{∗}-algebras B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) and the Ψ^{∗}-algebra Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is now straightforward. The short exact sequence (83) leads to the following 6-term exact sequence in K-Theory:

\begin{matrix}ℤ & ⟶ & K_{0}(B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )) & ⟶ & K_{0}(C(C_{de}X)) \\ ∂↑ & & & & ↓∂ \\ K_{1}(C(C_{de}X)) & ⟵ & K_{1}(B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )) & ⟵ & 0 \end{matrix}  (84)

where we have used dim : K_{0}(K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ))) ⟶^{≅} ℤ, and the isomorphism K_{1}(K(ϱN^{𝔟}L^{2}(X,^{de}Ω^{\frac{1}{2}} ))) = 0. The boundary map ∂ : K_{1}(C(C_{de}X)) ⟶ ℤ is simply the index map, i.e. we have ∂[f] = ind a if a ∈ M_{L}(B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )) satisfies τ_{de}^{(𝔟)}(a) = f ∈ M_{L}(C(C_{de}X))^{−1}, i.e. ∂ can be computed using the index formula in Theorem 5.17. By Theorem 5.18 we know that ∂ is onto which immediately implies the following result by the exactness of (84). 6.7 Proposition. Let 𝔟 ∈ ℝ be arbitrary. Then we have:

K_{0}(B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )) ≅ K_{0}(C(C_{de}X)) ≅ K^{0}(C_{de}X) and
K_{1}(B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} )) ≅ N(∂ := ind : K^{1}(C_{de}X) ⟶ ℤ) .

Since Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) is a Ψ^{∗}-algebra in the C^{∗}-algebra B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) by Theorem 6.4, it is closed under the holomorphic functional calculus, hence, the K-Theory of Ψ_{de}^{0,0}(X,^{de}Ω^{\frac{1}{2}} ) and B^{(𝔟)}(X,^{de}Ω^{\frac{1}{2}} ) coincide by [5, Appendix] – see also [2, Théorème 1.3.1] or [18, p. 109].

7 Wodzicki-residue and Dixmier-trace for

double-edge operators

Throughout this section let N := dim X = 1 + m + n be the dimension of X. We are going to show that Dixmier-trace and Wodzicki-residue coincide on an ideal containing the space Ψ_{de}^{−N,−∞}(X,^{de}Ω^{\frac{1}{2}} ). For simplicity, we restrict ourselves to sections of the double-edge half density bundle ^{de}Ω^{\frac{1}{2}} . The results, however, clearly extend to the case of operators acting on sections of finitedimensional vector bundles. Let us first recall some basic facts.

Review on the Dixmier trace

For a compact operator T : H_{1} ⟶ T_{2} between the separable Hilbert spaces H_{1} and H_{2}, let (μ_{j}(T))_{j∈ℕ_{0}} be the sequence of singular values of T, i.e.

μ_{j}(T) = λ_{j}(\sqrt{T^{∗}T}) = inf \{∥T − F ∥_{L(H_{1},H_{2})} : F ∈ L(H_{1}, H_{2}) , dim R(F) ≤ j\} ,

where (λ_{j}(\sqrt{T^{∗}T}))_{j∈ℕ_{0}} is the sequence of eigenvalues of the positive, compact operator \sqrt{T^{∗}T} ∈ L(H_{1}) in decreasing order and with multiplicities. Then the space

ℓ^{1,∞}(H_{1}, H_{2}) := \{T ∈ K(H_{1}, H_{2}) : ∥T ∥_{ℓ^{1,∞}} := sup_{N≥2} \frac{1}{log N} ∑_{j=0}^{N−1} μ_{j}(T) < ∞\}

is a Banach space with respect to the norm ∥ · ∥_{ℓ^{1,∞}}, and for A ∈ L(H_{2}, H_{3}), B ∈ L(H_{0}, H_{1}), and T ∈ ℓ^{1,∞}(H_{1}, H_{2}) we have AT B ∈ ℓ^{1,∞}(H_{0}, H_{3}) with ∥AT B∥_{ℓ^{1,∞}} ≤ ∥A∥_{L(H_{2},H_{3})}∥T ∥_{ℓ^{1,∞}}∥B∥_{L(H_{0},H_{1})}, where H_{0} and H_{3} are arbitrary, separable Hilbert spaces; thus, ℓ^{1,∞} is an operator ideal that is called the Dixmier ideal. Clearly, we have ℓ^{1}(H_{1}, H_{2}) ⊆ ℓ^{1,∞}(H_{1}, H_{2}) where

ℓ^{1}(H_{1}, H_{2}) := {T ∈ K(H_{1}, H_{2}) : (μ_{j}(T))_{j∈ℕ_{0}} ∈ ℓ^{1}(ℕ_{0})}

is the ideal of trace class operators. Our basic example is the following: 7.1 Proposition. For any closed manifold Y of dimension n, the embedding H^{n}(Y, Ω^{\frac{1}{2}} ) ⟶ L^{2}(Y, Ω^{\frac{1}{2}} )) belongs to the Dixmier ideal ℓ^{1,∞}(H^{n}, L^{2}).

The operator ideal ℓ^{1,∞} is important because for any fixed, separable Hilbert space Dixmier [8] constructed a family of traces

Tr_{ω} : ℓ^{1,∞}(H) := ℓ^{1,∞}(H, H) ⟶ ℂ

vanishing on the ideal ℓ^{1}(H) of trace class operators on H. Here, ω stands for an averaging procedure that is used to extend T ⟼ lim_{N→∞} \frac{1}{log N} ∑_{j=0}^{N−1} μ_{j}(T) to ℓ^{1,∞}(H). Moreover, the trace Tr_{ω} is bounded with respect to ∥ · ∥_{ℓ^{1,∞}}, i.e. we have |Tr_{ω}(a)| ≤ ∥a∥_{ℓ^{1,∞}}. For more details see [7].

Wodzicki-residue

It is well-known – see for instance [4], [15], or [44] – that up to a multiplicative constant there is only one trace on the algebra Ψcl^{ℤ}(Y, Ω^{\frac{1}{2}} ) := ⋃_{m∈ℤ} Ψ_{cl}^{m}(Y, Ω^{\frac{1}{2}} ) of classical pseudodifferential operators on a closed manifold Y; moreover, it vanishes on the ideal of smoothing operators Ψ^{−∞}(Y, Ω^{\frac{1}{2}} ). If n is the dimension of Y, it is given by

W-Res : Ψcl^{ℤ}(Y, Ω^{\frac{1}{2}} ) ⟶ ℂ : a ⟼ ∫_{S^{∗}Y} a_{−n}ı_{R}ω^{n} ,

and is called the Wodzicki or non-commutative residue. Here a_{−n} ∈ S^{[−n]}(T^{∗}Y) is the term of homogeneity −n in the homogeneous expansion of the full symbol a + Ψ^{−∞}(Y, Ω^{\frac{1}{2}} ) ∈ Ψcl^{ℤ}(Y, Ω^{\frac{1}{2}} )/Ψ^{−∞}(Y, Ω^{\frac{1}{2}} ) of a, and ı_{R}ω^{n} is the contraction of the n-th power of the symplectic form on T^{∗}Y with the radial vector field R in the fibers.

Note the remarkable fact that the component a_{−n} depends on choices but the Wodzicki residue W-Res(a) does not. For more details we refer the reader to [4], [11], [15], or [44].

By Proposition 7.1 pseudodifferential operators in Ψ_{cl}^{−n}(Y, Ω^{\frac{1}{2}} ) belong to ℓ^{1,∞}(L^{2}(Y, Ω^{\frac{1}{2}} )), and the Dixmier trace Tr_{ω} is well-defined on them. The following result is due to Connes [6].

7.2 Theorem. There exists a constant c_{n} ∈ ℂ such that we have for all a ∈ Ψ_{cl}^{−n}(Y, Ω^{\frac{1}{2}} )

Tr_{ω}(a) = c_{n}W-Res(a) .

In particular, Tr_{ω}(a) does not depend on the choice of the averaging procedure ω.

Wodzicki-residue for double-edge operators

For simplicity, let Ψ_{de}^{ℤ,−∞}(X,^{de}Ω^{\frac{1}{2}} ) := ⋃_{m∈ℤ} Ψ_{de}^{m,k}(X,^{de}Ω^{\frac{1}{2}} ). It is shown in [22] that there is up to a constant factor a unique trace on Ψ_{de}^{ℤ,−∞}(X,^{de}Ω^{\frac{1}{2}} ) that vanishes on the residual ideal Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ). For completeness, let us sketch the proof. Exactly as in [34], we see that the spectral sequence associated to the filtration by order of the Hochschild complex of I_{σ} := Ψ_{de}^{ℤ,−∞}/Ψ_{de}^{−∞,−∞} degenerates at E^{2} with E_{p,q}^{2} ≅ 0 for q ≠ N and E_{p,N}^{2} ≅ H_{rel}^{N−p}(^{de}S^{∗}X × S^{1}). This implies HH_{k}(I_{σ}) ≅ H_{rel}^{2N−k}(^{de}S^{∗}X × S^{1}), hence by the Künneth formula dim HH^{0}(I_{σ}) = dim HH_{0}(I_{σ}) = 1, and there is up to a constant a unique trace on I_{σ}. It is given by restricting the Wodzicki residue on the closed manifold 2X, the double of X, to Ψ_{de}^{ℤ,−∞}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ Ψ^{ℤ}(2X, Ω^{\frac{1}{2}} ), i.e. we have

W-Res_{de} : Ψ_{de}^{ℤ,−∞}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ℂ : a ⟼ ∫_{^{de}S^{∗}X} a_{−N}ı_{R}ω_{de}^{N} ,  (85)

where a_{−N} ∈ S^{[−N]}(^{de}T^{∗}X) is the part of homogeneity −N in the homogeneous asymptotic expansion of the full symbol a + Ψ_{de}^{−∞,−∞}(X,^{de}Ω^{\frac{1}{2}} ) ∈ I_{σ}, and ı_{R}ω_{de}^{N} denotes the contraction of the N-th power of the symplectic form ω_{de} on ^{de}T^{∗}X with the radial vector field in the fibers.

By (3), W-Res_{de}(a) consists of a finite sum of integrals of the form

∫_{ℝ_{z}^{m}} ∫_{ℝ_{y}^{n}} ∫_{0}^{∞} ∫_{S_{(ξ,η,ζ)}^{n+m}} a_{−N}(x, y, z, ξ, η, ζ)i_{ξ∂_{ξ}+η∂_{η}+ζ∂_{ζ}}(dξ ∧ dη ∧ dζ)\frac{dx}{x^{2+2n+m}} dy dz

with a_{−N} ∈ Ċ_{c}^{∞}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m})ˆ⊗_{π}S^{[−N]}(; ℝ_{ξ} × ℝ_{η}^{n} × ℝ_{ζ}^{m}). Therefore, we see that for any k > 1 + 2n + m the formula (85) continues to make sense for all a ∈ Ψ_{de}^{ℤ,−k}(X,^{de}Ω^{\frac{1}{2}} ) := ⋃_{m∈ℤ} Ψ_{de}^{m,−k}(X,^{de}Ω^{\frac{1}{2}} ), and we can extend the Wodzickiresidue to a map

W-Res_{de} : Ψ_{de}^{ℤ,−k}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ ℂ  (86)

which is still a trace for any such k.

Dixmier trace for double-edge operators

First we have to find which double-edge operators actually belong to the Dixmier ideal. As in the closed case, the answer to this question is split into an embedding and a boundedness result. In fact, the essential computation is local, hence we need the following local variants of the double-edge Sobolev spaces of Section 4. Let H_{de}^{0}(\overline{ℝ}_{+} ×ℝ_{y}^{n} ×ℝ_{z}^{m}) = L_{de}^{2}(\overline{ℝ}_{+} ×ℝ_{y}^{n} ×ℝ_{z}^{m}) be the Hilbert space of all measurable functions f : \overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m} ⟶ ℂ with

∥f∥_{L_{de}^{2}}^{2} := ∫_{ℝ_{z}^{m}} ∫_{ℝ_{y}^{n}} ∫_{0}^{∞} |f(x, y, z)|^{2} \frac{dx}{x^{2+2n+m}} dy dz < ∞ .

Moreover, for M ∈ ℕ_{0} and k ∈ ℝ let H_{de}^{M+1}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}) be the space of all f ∈ H_{de}^{M}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}) with x^{2}∂_{x}f, x^{2}∂_{y}f, x∂_{z}f ∈ H_{de}^{M}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}),

x^{k}H_{de}^{M}(\overline{ℝ}_{+}×ℝ_{y}^{n}×ℝ_{z}^{m}) := \{f ∈ C^{−∞}(\overline{ℝ}_{+}×ℝ_{y}^{n}×ℝ_{z}^{m}) : x^{−k}f ∈ H_{de}^{M}(\overline{ℝ}_{+}×ℝ_{y}^{n}×ℝ_{z}^{m})\} ,

and let these spaces be endowed with the obvious Hilbert space structure. 7.3 Lemma. For any k > 2N and any φ ∈ C_{c}^{∞}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}), the map

M_{φ} : x^{k}H_{de}^{N}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}) ⟶ L_{de}^{2}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}) : f ⟼ φf

belongs to the Dixmier ideal ℓ^{1,∞}(x^{k}H_{de}^{N}, L_{de}^{2}). Proof: Let ψ ∈ C^{∞}(ℝ_{+}) be with ψ(x) = 0 for x ≤ \frac{1}{2} and ψ(x) = 1 for x ≥ 1. For ℓ ∈ ℕ let ψ_{ℓ} : x ⟼ ψ(2^{ℓ}x). We are going to show

∥M_{φψ_{ℓ+1}} − M_{φψ_{ℓ}}∥_{ℓ^{1,∞}} ≤ C 2^{\frac{2+ℓ}{2} (2N−k)} for all ℓ ∈ ℕ .  (87)

Since ℓ^{1,∞} is complete, and we have M_{φψ_{ℓ}}f ⟶ M_{φ}f in L_{de}^{2}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}) for all f ∈ L_{de}^{2}(\overline{ℝ}_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}), the lemma follows.

For (87) it suffices to consider the following commutative diagram

[figure: x^{k}H_{de}^{N}; (T_{ℓ})^{∗}; M_{(ψ_{ℓ+1}−ψ_{ℓ})φ}; L_{de}^{2}; x^{k}H_{de}^{N}; M_{β}; L_{de}^{2}; (T_{ℓ}^{−1})^{∗}; L_{de}^{2}; M_{(ψ_{ℓ+1}−ψ_{ℓ})φ}]

where β ∈ C_{c}^{∞}(ℝ_{+} × ℝ_{y}^{n} × ℝ_{z}^{m}) satisfies β(x, y, z) = 1 for x ∈ [1, 4], therefore, M_{β} ∈ ℓ^{1,∞}(x^{k}H_{de}^{N}, L_{de}^{2}) by Proposition 7.1, T_{ℓ} is induced by x ⟼ 2^{−ℓ−2}x, and we have

∥(T_{ℓ})^{∗}∥_{L(x^{k}H_{de}^{N})}∥(T_{ℓ}^{−1})^{∗}∥_{L(L_{de}^{2})} ≤ C 2^{− \frac{2+ℓ}{2} (k−2N)}

by an easy induction. □

7.4 Proposition. The embedding ϱ_{N}^{k}H_{de}^{N}(X,^{de}Ω^{\frac{1}{2}} ) ⟶ L^{2}(X,^{de}Ω^{\frac{1}{2}} ) belongs to the Dixmier ideal ℓ^{1,∞}(ϱ_{N}^{k}H_{de}^{N}(X,^{de}Ω^{\frac{1}{2}} ), L^{2}(X,^{de}Ω^{\frac{1}{2}} )), provided k > 2N. Proof: By a partition of unity, the statement can be reduced to the local cases considered in Proposition 7.1 and Lemma 7.3. □

7.5 Corollary. For k > 2N, we have Ψ_{de}^{−N,−k}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ ℓ^{1,∞}(L^{2}(X,^{de}Ω^{\frac{1}{2}} )).

The following theorem generalizes the result of Connes (Theorem 7.2) to the double-edge case. 7.6 Theorem. There exists C_{N} ∈ ℂ such that for all k > 2N and all a ∈ Ψ_{de}^{−N,−k}(X,^{de}Ω^{\frac{1}{2}} ),

Tr_{ω}(a) = C_{N}W-Res_{de}(a) .  (88)

In particular, Tr_{ω}(a) is independent of the choice of the averaging procedure ω. Proof: First note that by (86) and Corollary 7.5, both sides of (88) are welldefined and have the trace property. We are going to apply Theorem 7.2.

Let δ > 0 be with k > 2N + δ. Choose φ ∈ C^{∞}(\overline{ℝ}_{+}) with φ = 0 for ϱ ≤ \frac{1}{2} and φ = 1 for ϱ ≥ 1, and let φ_{ε} := φ(εϱ_{N}) : X ⟶ \overline{ℝ}_{+}. Then the kernel of a_{ε} := φ_{ε}aφ_{ε} is supported in the interior of X^{2}, hence a_{ε} ∈ Ψcl^{ℤ}(2X, Ω^{\frac{1}{2}} ), and we have Tr_{ω}(a_{ε}) = C_{N}W-Res(a_{ε}) = C_{N}W-Res_{de}(a_{ε}) by Theorem 7.2 with a universal constant C_{N} independent of a_{ε}. On the other hand, we have

|Tr_{ω}(a_{ε}) − Tr_{ω}(a)| = |Tr_{ω} ((1 − φ_{ε}^{2})a)|
≤ ∥(1 − φ_{ε}^{2})ϱ_{N}^{δ}∥_{L(L^{2}(X,^{de}Ω^{\frac{1}{2}} ))}∥ϱ_{N}^{−δ}a∥_{ℓ^{1,∞}}
= ∥(1 − φ_{ε}^{2})ϱ_{N}^{δ}∥_{sup_{X}}∥ϱ_{N}^{−δ}a∥_{ℓ^{1,∞}}
⟶^{ε→0} 0

because of ϱ_{N}^{−δ}a ∈ Ψ_{de}^{−N,−k+δ}(X,^{de}Ω^{\frac{1}{2}} ) ⊆ ℓ^{1,∞}(L^{2}(X,^{de}Ω^{\frac{1}{2}} )) by Corollary 7.5 and the choice of δ.

Similarly, we see using the local presentation for W-Res_{de}

|W-Res_{de}(a_{ε}) − W-Res_{de}(a)| = |W-Res_{de}((1 − φ_{ε}^{2})a)|
≤ C(a)∥(1 − φ_{ε}^{2})ϱ_{N}^{δ}∥_{sup_{X}} ⟶^{ε→0} 0 .

This completes the proof. □

ACKNOWLEDGEMENTS

It is a pleasure to thank Richard Melrose for suggesting the study of the double-edge calculus, for his generous advice, and for many inspiring discussions influencing large parts of the paper. We are also indebted to Victor Nistor and Elmar Schrohe for useful remarks on Section 5 and Section 7. The remarks of the referee were helpful for improving parts of the manuscript.

The first author was supported by a scholarship of the German Academic Exchange Service (DAAD) within the Hochschulsonderprogramm III von Bund und Ländern. He wishes to thank the Massachusetts Institute of Technology, where the paper was written for the invitation and warm hospitality.

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